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Domain-perturbed problems for ordinary linear differential operators. Froese, John 1966

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DOMAIN-PERTURBED PROBLEMS FOR ORDINARY LINEAR DIFFERENTIAL OPERATORS ^ by JOHN FROESE B . A . U n i v e r s i t y of M a n i t o b a , 1952 M . A . Queen's U n i v e r s i t y , 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS' FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , . 1 9 6 6 . I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M a t h e m a t i c s T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a D a t e M a y 10, 1966  The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of JOHN FROESE B . A . , University of Manitoba, 1952 M . A . , Queen's University, 1961 Monday, May 9th, 1966, at 3:30 p.m. in Room 225, Mathematics Building COMMITTEE IN CHARGE Chairman: J . A . Jacobs Douglas Derry Earl D. Rogak Charlotte Froese C.A. Swanson Elod Macskasy Roy Westwick External Examiner: R.R.D. Kemp Queen's University Kingston, Ontario Research Supervisor: C.A. Swanson . PERTURBED PROBLEMS FOR ORDINARY LINEAR DIFFERENTIAL OPERATORS Abstract The v a r i a t i o n of the eigenvalues and eigenfunctions of an ordinary l i n e a r s e l f -adjoint d i f f e r e n t i a l operator L i s considered under perturbations of the domain of L. The basic problem i s defined as a suitable singular eigenvalue problem f o r L on the open i n t e r v a l m_ < a < u u + and i s assumed to have at le a s t one r e a l eigenvalue X of m u l t i p l i c i t y k. The perturbed problem i s a regular s e l f - a d j o i n t problem defined f o r L on a closed subinterval [a,b] of (CJO_,U)+). It i s proved under suitable conditions on the boundary operators of the perturbed ^problem that exactly k perturbed eigen-values -* X as a,b -• u ) _ , u u + . Further, asymptotic estimates are obtained f o r u ^ j -as a,b -• u J _ > u ) + . The other r e s u l t s are refinements which lead to asymptotic estimates fo r the. eigenf unctions and v a r i a t i o n a l formulae f o r the eigenvalues. The conditions on the l i m i t i n g behaviour of the boundary operators depend strongly on the nature of the s i n g u l a r i t i e s iw_.»<i)+. I f for some number lQ>l0 not an eigenvalue, l i n e a r l y independent solutions of Lx = IQX e x i s t which are asymptotically ordered at t u _ , then uu_ i s palled a class 1 s i n g u l a r i t y . In the case that both (i)_,u! + are class 1 s i n g u l a r i t i e s , very-general boundary operators permit the conver-gence of to X., Class 2 s i n g u l a r i t i e s are defined as follows: If a l l solutions of Lx = tQx are square-integrable on (ID , c] f o r any c s a t i s f y i n g w_ < c < ou+, then u>_ i s c a l l e d a class 2 s i n g u l a r i t y . An asymptotic ordering of the solutions i s not assumed i n t h i s case. Since the behaviour of the solutions of .. Lx = <tex i s e s s e n t i a l l y a r b i t r a r y when both oo_,uu+ are class 2 s i n g u l a r i t i e s , the generality of the boundary operators has to be s a c r i f i c e d to ensure that - X. Certain one end perturbation problems and examples also are considered. GRADUATE STUDIES Asymptotic Analysis Complex Variable Functional Analysis Measure and Integration Point Set Topology C. A. Swanson R.A. Cleveland A. H. Cayford S.W. Nash P. S. Bullen i i ABSTRACT The v a r i a t i o n o f the e i g e n v a l u e s and e i g e n f u n c t i o n s of an o r d i n a r y l i n e a r s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r L i s c o n s i d e r e d under p e r t u r b a t i o n s of the domain of L . The b a s i c  p r o b l e m i s d e f i n e d as a s u i t a b l e s i n g u l a r e i g e n v a l u e p r o b l e m f o r L on the open i n t e r v a l < s < u)+ and i s assumed t o have at l e a s t one r e a l e i g e n v a l u e X of m u l t i p l i c i t y k. The p e r t u r b e d  p r o b l e m i s a r e g u l a r s e l f - a d j o i n t p r o b l e m d e f i n e d f o r L on a c l o s e d s u b i n t e r v a l [ a , b ] of (uo ,^ci)+). I t i s p r o v e d , under s u i t a b l e c o n d i t i o n s on the boundary o p e r a t o r s of the p e r t u r b e d p r o b l e m t h a t e x a c t l y k p e r t u r b e d e i g e n v a l u e s -» X as a ,b -» «)_,uo+. F u r t h e r , a s y m p t o t i c e s t i m a t e s are o b t a i n e d f o r ^ab ~ s s & , ^ > ~* UL-'U U+' ^ n e ° t h e r r e s u l t s are r e f i n e m e n t s w h i c h l e a d t o a s y m p t o t i c e s t i m a t e s f o r the e i g e n f u n c t i o n s and v a r i a t i o n a l f o r m u l a e f o r the e i g e n v a l u e s . The c o n d i t i o n s on the l i m i t i n g b e h a v i o u r of the boundary o p e r a t o r s depend s t r o n g l y on the n a t u r e of the s i n g u l a r i t i e s a ) _ 3 « ) + . I f f o r some number iQ, lQ not an e i g e n v a l u e , l i n e a r l y independent s o l u t i o n s of Lx = iQx e x i s t w h i c h are a s y m p t o t i c a l l y o r d e r e d at «u_, t h e n uu_ i s c a l l e d a c l a s s 1 s i n g u l a r i t y . In the case t h a t b o t h u ) _ , « ) + are c l a s s 1 s i n g u l a r i t i e s , "very "genera l boundary o p e r a t o r s p e r m i t the convergence of t o x. C l a s s 2 s i n g u l a r i t i e s are d e f i n e d as f o l l o w s : I f a l l s o l u t i o n s of Lx = ^ 0 x are s q u a r e - i n t e g r a b l e on (u> , c ] f o r any c s a t i s f y i n g t»_ < c < di , t h e n t» i s c a l l e d a c l a s s 2 s i n g u l a r i t y . An i i i a s y m p t o t i c o r d e r i n g of the s o l u t i o n s i s not assumed i n t h i s case. S i n c e the b e h a v i o u r o f t h e - s o l u t i o n s o f Lx = t Q x i s e s s e n t i a l l y a r b i t r a r y when b o t h u u _ . > < » + are c l a s s 2 s i n g u l a r i t i e s , the g e n e r a l i t y of t h e boundary o p e r a t o r s has t o be s a c r i f i c e d t o ensure t h a t X. C e r t a i n one end p e r t u r b a t i o n problems and examples a l s o a re c o n s i d e r e d . i v TABLE OF CONTENTS page INTRODUCTION ± CHAPTER I A s y m p t o t i c E s t i m a t e s f o r uo ,ai C l a s s 1 S i n g u l a r i t i e s 1. D e s c r i p t i o n of the b a s i c and p e r t u r b e d problems 6 2. Comparison of the b a s i c and p e r t u r b e d problems 7 3. U n i f o r m e s t i m a t e s f o r e i g e n f u n c t i o n s on [ a , b ] 21 4. A s y m p t o t i c v a r i a t i o n a l f o r m u l a e f o r e i g e n v a l u e s 35 5. The second o r d e r c a s e : w_>t« + l i m i t p o i n t s i n g u l a r i t i e s 38 6. One end p e r t u r b a t i o n problems 42 7. C l a s s 1 s i n g u l a r problems f o r w h i c h a l l b a s i c s o l u t i o n s are i n H 48 8. A one-end p e r t u r b a t i o n p r o b l e m ; a l l b a s i c ! s o l u t i o n s i n H(uo_,b] 60, 9. The second o r d e r c a s e ; oo ,m, c l a s s 1 l i m i t c i r c l e — + s i n g u l a r i t i e s 62 CHAPTER I I C l a s s 2 S i n g u l a r Problems P r e l i m i n a r i e s ~ 66 10. D e s c r i p t i o n of the b a s i c and p e r t u r b e d problems 67 11. Comparison of the b a s i c and p e r t u r b e d problems 69 12. U n i f o r m e s t i m a t e s and a s y m p t o t i c v a r i a t i o n a l f o r m u l a e ' 73 13. One end p e r t u r b a t i o n problems 76 V page 14. The second o r d e r c a s e ; IM ,UU, c l a s s 2 l i m i t — -t-c i r c l e s i n g u l a r i t i e s 34 CHAPTER I I I Examples 1 5 - P r e l i m i n a r y remarks and lemmas 8 8 1 6 . The m o d i f i e d Hermite o p e r a t o r 8 9 17. Example 2 9 8 1 8 . Example 3 1 0 5 BIBLIOGRAPHY 1 0 8 ACKNOWLEDGEMENT , I w i s h t o express my g r a t i t u d e to"my s u p e r -v i s o r , D r . C. A. Swanson, f o r s u g g e s t i n g the' 1 t o p i c of t h i s t h e s i s , and f o r p r o v i d i n g v a l u a b l e guidance t h r o u g h o u t my graduate s t u d i e s and d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . T a l s o w i s h t o thank D r . R . C . R i d d e l l f o r h i s c o n s t r u c t i v e c r i t i c i s m of the d r a f t f o r m of t h i s work. The generous f i n a n c i a l suppor t of the U n i v e r -s i t y of B r i t i s h C o l u m b i a and the N a t i o n a l R e s e a r c h C o u n c i l of Canada i s g r a t e f u l l y acknowledged. 1. INTRODUCTION Eigenvalue problems w i l l be considered for- t;he n-th order, ordinary, l i n e a r d i f f e r e n t i a l operator L defined .by n ( C D L X « y ~ V ( S ) x ( n - i } ( S ) 1=0 on the open i n t e r v a l u>_ < s < «>+ where k and p^, i = 0,1,...,n are real-valued functions on t h i s i n t e r v a l with the properties that: ( i ) V±(a) e C n" i( t t,_, t t» +) a ; i - 0,1,...,n ; ( i i ) k(s) i s piecewise continuous on (<0_><«>+) J and ( i i i ) p (B) / 0 and k ( s ) > 0 on Furthermore we assume that the operator v k ( s).Lx = £ p ( s ) x(n-lk») i=0 1 i s ' f o r m a l l y s e l f - a d j o i n t , i . e . , k ( s)«Lx coincides with i t s Lagrangian adjoint [k(s)« Lx]"*" where n (0.2) C k ( s ) . L x ] + - J ~ | ( - D n " i [ p 1 x ] ( n " i ) . The points w. and ^ are i n general s i n g u l a r i t i e s of L; the p o s s i b i l i t y that they are .+ « i s not excluded. It w i l l be convenient to use the following notations: 2. (0.3) (x>y)t = J x(u)y(u)k(u)du , W _ _< S < t _< uj ; s U) ( x , y ) a - ( x , y ) a + • ( x , y ) b = ( x , y ) b ; it) ( x , y ) = ( x ^ y ) ^ ; n (0 .4 ) [ x y ] ( 8 ) = Y2 Y2 ( - D ^ ( k ) ( s ) [ p n . m ( s ) y ( s ) j ( ^ ; m=l j+k=m-l j > 0 , k>0 [xy](a> ) =- l i m [xy ] ( s ) — S -* UJ + S i n c e the o p e r a t o r k« L i s f o r m a l l y s e l f - a d j o i n t G r e e n ' s sym-m e t r i c f o r m u l a has the f o r m (0.5) ( L x , y ) g - ( x , L y ) * = [ x y ] ( t ) - [ x y ] ( s ) . L e t H , H [ a , b ] denote the H i l b e r t spaces w h i c h are the Lebesgue spaces w i t h r e s p e c t i v e i n n e r p r o d u c t s ( x , y ) , ( x , y ) Q and norms 1 1 ||x|J = ( x 3 x ) ^ , )]x||^  = i(x,x)lf , < a < b < u i + . F o r c any i n t e r m e d i a t e p o i n t , u/_ < c < uu+, we l i k e w i s e d e f i n e H(m_,c] , H [ c , w + ) t o be the H i l b e r t spaces w h i c h are the Lebesgrue spaces w i t h r e s p e c t i v e i n n e r p r o d u c t s ( x , y ) c , ( x , y ) and norms Hx|J = [ ( x , x ) ] , ||x||c = [ ( x , x ) c ] . From (0.5) i t i s c l e a r t h a t '[xy]|u } (or [xy3(u>_)) e x i s t s p r o v i d e d x , y , L x , and Ly are i n H[c ,ou + ) (or x , y , L x and Ly are i n H( a i _ , c j ) . L e t a Q and b Q be f i x e d numbers s a t i s f y i n g oi_ < a Q < b Q < uo+ and l e t R Q be the r e c t a n g l e i n the a - b - p l a n e d e s c r i b e d by the 3. i n e q u a l i t i e s u>_ < a <_ a Q , b Q _< b < uu+ . E v e r y c l o s e d , bounded i n t e r v a l [ a , b ] , tu_ < a _< a Q , b Q _< b < t » + , can be a s s o c i a t e d i n a , o n e - t o - o n e manner w i t h a p o i n t of R Q . F o r e v e r y such [ a , b ] we s h a l l c o n s i d e r the e i g e n v a l u e p r o b l e m (0.6) Ly = |iy , l £ y = 0, i = 1 , 2 , . . . , m , U^y = 0, i = 1,2,...,n-m where and are the l i n e a r boundary o p e r a t o r s n-1 (0.7) f U^y = 5 1 < * l k ( a ) y ( k ) ( a ) ., i = l , 2 , . . . , m k=0 n-1 U b y = X P l k ( b ) y ( k ) ( b ) ^ i = l , 2 , . . . , n -k=0 m where y a ^ k ( a ) , ^ ^ ( k ) a r e r e a l - v a l u e d f u n c t i o n s on the r e s p e c -t i v e i n t e r v a l s uu_ < a _< a Q , b Q _< b < uu+ and are such t h a t the c o n d i t i o n s IT%- = U-Py = 0 , i = l , 2 , . . ; , m , j = 1,2,...,n-m form a l i n e a r l y i n d e p e n d e n t s e l f - a d j o i n t se t o f boundary c o n d i t i o n s f o r L (See [ 3 ] , p p . 2 8 8 - 291). Our p r o b l e m i s t o o b t a i n e s t i m a t e s f o r each e i g e n v a l u e tj = ia ^ of (0.6) f o r a , b near « ) + under h y p o t h e s i s t h a t w i l l ensure t h a t the l i m i t s of l i a ^ as a , b -* u) , i u + w i l l e x i s t . A c c o r d i n g l y , we s h a l l assume t h a t e i g e n v a l u e s X of s u i t a b l e s i n g u l a r e i g e n v a l u e problems f o r L on (cu_,w +) e x i s t . I f the e i g e n s p a c e of X i s k - d i m e n s i o n a l our f i r s t theorem shows i n p a r t i c u l a r t h a t at l e a s t k e i g e n v a l u e s of (0.6) converge t o X as a , b - ou ,co, ". Our o t h e r r e s u l t s are — " T r e f i n e m e n t s of t h i s w h i c h l e a d t o a s y m p t o t i c e s t i m a t e s f o r e i g e n f u n c t i o n s . The method of e s t i m a t i o n used here i s due t o H . F . B o h n e n b l u s t (see [ l l ] , p . 1553). 4. R e s u l t s l i k e these have "been p r e v i o u s l y obtained f o r the second order, ord i n a r y case by C.A. Swanson. In [ 1 1 J he considers the case when, i n Weyl's c l a s s i f i c a t i o n , both oi_ and tt)_^ are l i m i t c i r c l e s i n g u l a r i t i e s , t.and i n [13] he considers the cases ( i ) when both tt>_ and u>+ are l i m i t p o i n t s i n g u l a r i t i e s ; ( i i ) when uu i s a l i m i t c i r c l e s i n g u l a r i t y and uo+ i s a l i m i t p o i n t s i n g u l a r i t y . Swanson makes strong use of two well-known theorems of Weyl ( [ 9 ] J P ' 3 5 and p. 4 5 ) i n s e t t i n g up s u i t a b l e s i n g u l a r eigenvalue problems. However, f o r higher order cases (n > 2 ) these theorems are no longer v a l i d so that assumptions w i l l have to be made i n p a r t i c u l a r about the behaviour of the s o l u t i o n s of Lx = i x , Im t ^ 0, at u>_ and at uu+ . Also we w i l l always r e q u i r e the existence of at l e a s t one r e a l eigenvalue f o r the s i n g u l a r eigenvalue problem on (t*>_,uo+) at hand. I t i s e a s i l y seen i n [ l l ] and [ 1 3 ] that the " l i m i t p o i n t , l i m i t c i r c l e " c l a s s i f i c a t i o n of s i n g u l a r i t i e s f o r the second order d i f f e r e n t i a l operator i s not a n a t u r a l c l a s s i f i c a t i o n i n r e l a t i o n to domain-perturbed problems. Moreover these terms " l i m i t p o i n t , l i m i t c i r c l e " have l i t t l e or no meaning i n reference t o s i n g u l a r i -t i e s of higher order d i f f e r e n t i a l operators. Consequently, we s h a l l use the f o l l o w i n g c l a s s i f i c a t i o n of the s i n g u l a r i t i e s tu_ and U D + of L; l e t l Q be any complex number, Im lQ £ 0 , and l e t cpi, i - 1 , 2 , ...,n, be l i n e a r l y independent s o l u t i o n s ( h e r e a f t e r t o be r e f e r r e d to as b a s i c s o l u t i o n s ) of L Q x = 0 where L Q= L - lQ . I f there e x i s t b a s i c s o l u t i o n s cpi, i = l , . . . , n such that ( 0 . 8 ) l i m ®j(s) = o or » s - U>+ Cpj ( s ) f o r each p a i r ep., cp i , j = l , 2 , . . . , n , i ^ j , then u> w i l l be 5. r e f e r r e d t o as c l a s s 1 s i n g u l a r i t i e s . Note t h a t i n t h i s case iu + cannot be a c c u m u l a t i o n p o i n t s of z e r o s f o r the b a s i c s o l u t i o n s . I t w i l l be seen i n chapter 1 t h a t f o r t h i s case o n l y s l i g h t r e s t r i c t i o n s are needed on the l i m i t i n g b e h a v i o u r of the boundary o p e r a t o r s ( 0 . 7 ) t o o b t a i n convergence o f the e i g e n v a l u e s of ( 0 . 6 ) as a , b -» (w_,uu+ (see ( 2 . 7 ) - ( 2 . 1 2 ) ) . On the o ther hand u>_ (or ti)+) w i l l be c a l l e d a c l a s s 2 s i n g u l a r i t y when the b e h a v i o u r o f the b a s i c s o l u t i o n s i s e s s e n t i a l l y a r b i t r a r y as s «)_ (or s - uo +). In p a r t i c u l a r t h i s i n c l u d e s cases f o r w h i c h ( 0 . 8 ) does not h o l d or f o r which the b a s i c s o l u t i o n s o s c i l l a t e i n -f i n i t e l y o f t e n as s -• uu__ (or s - u) + ) . I t w i l l be seen i n c h a p t e r 2 t h a t i n t h i s case more r e s t r i c t i v e c o n d i t i o n s are, needed on the l i m i t i n g b e h a v i o u r o f the boundary o p e r a t o r s ( 0 . 7 ) as a -» oi (or b - o)+) t o o b t a i n convergence of the e i g e n v a l u e s of ( 0 . 6 ) (see ( 1 0 . 3 ) ) . The s i n g u l a r i t i e s ua (or w+) w i l l be f u r t h e r c h a r a c t e r i z e d by the number of b a s i c s o l u t i o n s • t h a t are i n E(m_,c] (or i n H[c ,u) , ) ) where c i s any number s a t i s f y i n g u) < c < ID , . Thus + — + f o r n = 2 , UJ_ i s a " l i m i t c i r c l e " s i n g u l a r i t y i f b o t h b a s i c s o l u -t i o n s are i n H(u) >c] ; o t h e r w i s e uu_ i s a " l i m i t p o i n t " s i n g u l a r i t y . The p l a n i s as f o l l o w s : Chapter I w i l l be devoted t o p e r t u r b a t i o n problems where b o t h iu_ and cu+ are c l a s s 1 s i n g u l a r i -t i e s . I n Chapter I I p e r t u r b a t i o n problems are c o n s i d e r e d f o r w h i c h b o t h iu_ and uo+ a re c l a s s 2 s i n g u l a r i t i e s and a l l the b a s i c s o l u t i o n s are i n H . F i n a l l y examples of p e r t u r b a t i o n problems w i l l be g i v e n i n Chapter I I I w h i c h w i l l i l l u s t r a t e the m a t e r i a l i n C h a p t e r s I and I I . CHAPTER I ASYMPTOTIC ESTIMATES, FOR PROBLEMS WITH u)_,0) CLASS 1 SINGULARITIES ! • D e s c r i p t i o n of the b a s i c and p e r t u r b e d p r o b l e m s . One type of s i n g u l a r p r o b l e m on (u>_.jU0+) t o be c o n s i d e r e d i n t h i s c h a p t e r i s the case t h a t no s o l u t i o n o f L Q x = 0 i s i n H. More p r e c i s e l y we s h a l l assume the e x i s t e n c e of b a s i c s o l u t i o n s tp^cpg, . . . ,<pn such t h a t f o r any number c, ui < c < uo+; (1) cpj € H [ c , a u + ) , cpj 4 H ( u u _ , c ] , j = 1,2,. . . ,,m, qpj e H(u)_,c] , cp^  4 H [ c , t « + ) , j = m + l , . . . , n ; (1.1) ( i i ) l i m x T a . = 0 , i = l , 2 , . . . , m - l , a - u)_ ^ i ^ ' « P ± ( b ) l i m r , v = 0 , i = m + l , . . . , n - l . b - o>+ ' i + l ^ L e t D be the set of a l l x € H such t h a t x e C (<*>_. .»u>+) and x^"*"^ i s a b s o l u t e l y c o n t i n u o u s on e v e r y c l o s e d bounded sub-I n t e r v a l of (oi_,oo +). Then the s i n g u l a r e i g e n v a l u e p r o b l e m on (w_, © + ) (1.2) Lx = \ x , x e D i s c a l l e d the b a s i c p r o b l e m . Our main assumption i s t h a t t h e r e e x i s t s at l e a s t one e i g e n v a l u e \ of t h i s p r o b l e m . In [ 6 ] , K. K o d a i r a shows t h a t the d i m e n s i o n of the s o l u t i o n space of Lx = lx i n H i s independent of l , p r o v i d e d o n l y t h a t Im I £ 0. S i n c e Lx = l x has no s o l u t i o n s In H , i t f o l l o w s t h a t a l l o e i g e n v a l u e s of (1.2) are n e c e s s a r i l y r e a l . In p a r t i c u l a r , l 7. i s not an e i g e n v a l u e and X i s n e c e s s a r i l y r e a l . F o r each [ a , b ] € R , l e t D [ a , b ] denote the se t o f a l l y € H [ a , b ] w h i c h s a t i s f y the f o l l o w i n g c o n d i t i o n s : ( i ) y. € C n " 1 [ a , b ] J y (n-1) i s a b s o l u t e l y c o n t i n u o u s on [ a , b ] ; ( i i ) Ly € H [ a , b ] j and ( i i i ) y s a t i s f i e s the homogeneous boundary c o n d i t i o n s of (0.6). Then the set D [ a , b ] w i l l be r e f e r r e d t o as the p e r t u r b e d domain and the e i g e n v a l u e p r o b l e m f o r (1.3) i s s e l f - a d j o i n t , i t f o l l o w s f rom the b o u n d a r y - f o r m f o r m u l a ( [ 3 ] , p . 2bb) and (0.5) t h a t the p e r t u r b e d p r o b l e m (1.3) i s s e l f - a d j o i n t . I t i s w e l l - k n o w n ( [ 3 h chapter 7) t h a t f o r such a s e l f - a d j o i n t p r o b l e m t h e r e e x i s t s a c o u n t a b l e se t of r e a l e i g e n v a l u e s a c c u m u l a t i n g o n l y at » and a se t of ( r e a l ) e i g e n -f u n c t i o n s complete i n H [ a , b J . 2. Comparison o f the b a s i c and p e r t u r b e d p r o b l e m s . To o b t a i n convergence of the e i g e n v a l u e s of (1.3) t o t h o s e of (1 .2) , r e s t r i c t i o n s w i l l have t o be imposed on the (1 .3) Ly = > y e D [ a , b J as the p e r t u r b e d p r o b l e m . ' S i n c e the se t of boundary c o n d i t i o n s L e t A ( a , b ) denote the n - b y - n m a t r i x (A. . ( a , b ) ) where 8. The symbols ( i ^ i ^ a n d (j'l'J'k^ w 1 1 1 b e referred to as sequences and w i l l represent any increasing sequences o f . k integers selected from the set {l , 2,...,n}. We adopt the following notations: ( 2 . 1 ) n a ( j r j m ) nb(<VJ"n-m) n a = det (^qpj_ ) , i , k = 1 , 2 , . . . ,m; = det (UjV- ) , i , k = 1 , 2 , . . . , n - m ; = h (l,m); n b = \ ( m + 1 * n ) » For x any normalized eigenfunction of ( 1 . 2 ) and for any sequence ( i n , i -| ) we l e t 6 ( i n , i . . n ) denote the determinant of the matrix v 1 m-l a v 1 m-l . ^ % • • • i. m-l . . . • • • - " ^ V i ^ -. . . ^ m-1 S i m i l a r l y for any-sequence U ^ n - m - l ) we l e t 8 b ( J n _ m - 1 ) denote the determinant of the matrix U b X ^ j ^ j n-m-1 n-m-1 n-m-1 9. The following notations w i l l be used: qp.(a) q>4_(b) ( 2 . 3 ) S a(i*d) = o a ( i , J ) U C P . ^ ; 5b(i»J) = a b ( i , j )ll<Pjllb; (2-*> < ( W i ) - 5 a ( l n j l m " l ) H ^ H a ; a • ^ M i ) - 6 b ( J l j J n - m - l ) ,,b . ^b ^ l ^ n - m - l ' ' " h^ > i * n ) a, •0 J*M 1- ) - ^ l ^ n - m * ' „ ,.b . ^b ^ J l i J n - m J " n b 11 V ' («_ < a _< a Q , b Q <, b < u) +, i = l ,2,...,m, j* = m+1, n. F i n a l l y for X an eigenvalue of ( 1 . 2 ) and A^ the corresponding eigenspace of dimension k we l e t ( 2 . 6 ) = sup |-0a(2,m)| , -a «= sup |^J(m+l,n-l)| a xeAx a D xeA x D 11x11=1 Hx||=l The assumptions below turn out to be s u f f i c i e n t to obtain con-vergence of the eigenvalues of ( 1 . 3 ) to those of ( 1 . 2 ) as a,b -» uo , o i , . 10. Assumptions; (a) The s i n g u l a r i t i e s tw_ and t»+ are not accumulation points of the zeros of tpjjj = 1,2, ...,n. (b) There exists a p o s i t i v e continuous function h(s) defined on («_,«),) such that l im h(s) = + » and such that the following + conditions are s a t i s f i e d : (2.7) ( i ) h(a) 5 a ( i , J ) = o ( l ) as a - cu_ and (2,b) h(b) §b(i,«j) - o ( l ) as b - » + f o r i = 1,2,...,m, j = m+l,...,n; ( i i ) There exi s t integers i Q , j where 1 _< i Q jC m, m+1 _< ' j _< n such that to *\ ^ W ^ i ^  ( 2 ' 9 ) h(a) \ * A ± ) i s bounded on (<«_,a0] f o r i =» 1,2,. .. ,m and f o r a l l sequences ( i . ^ i m ) f o r which i ^  l f c , k--= 1,2,...,m; and n b ( j 1 , j n _ m ) e p , ( b ) (2.10) b 1 n-m J h(b) h b <p, (b) o i s bounded on [b 0,t» +) for j = m+l,...,n and a l l sequences (Jl'j'n-m) for which j ^  J f c , k = m+l,...,n. (2.11) (C) ^ W x ) = 0[^(2,m)] = 0(-0&) - o ( l ) as a -» «)_ for i =» 1,2,... ,m and a l l sequences ( i i ^ ^ . i ) f o r which i ^  i ^ , k = 1,2,...,m-1; and (2.12) J e b ^ l ^ n - m - l ) " 0[ ^ ( w l , n - l ) ] - 0(4^) - o ( l ) 11. as b -» oo, f o r j = m+1,. . . ,n and a l l sequences ( d i > d n H t t l _ i ) f o r which j £ J , k = 1,2,...,n-m-l. (2.13) (2.14) (2.15) (2.16) These assumptions imply the f o l l o w i n g when a -• uu_,b -* uu h ( a ) o a ( i , j ) = o ( l ) , h ( b ) a b ( l , J ) = o ( l ) , I = 1,2,...,m, j = m+l,...,n; Tl ( I T i i ) a v 1 my + h a = o ( l ) f o r ( i - ^ i j t (l,m)j = o ( l ) f o r ( d - ^ d ^ ) t (m+l,n) J 1 * a T Hf (2.17) -©b ( J i > J " n _ m ) = °(l) f o r J = m+l,...,n and a l l sequences f o r which j / k=l,2,...,n-m. Also (2.9) and (2.10) imply that there e x i s t s neighbourhoods (uu_,aQ] , [bQ,u) ) of uu_, uo+ r e s p e c t i v e l y and a constant C such that a v 1 m7 ( i ^1 ) = o ( l ) f o r i = 1 , 2 , . . . , m and a l l sequences ( i 1 , i m ) f o r w h i c h ' i / i f c , k = l , 2 , . . . , m j (2.1») h a < C | o a ( i , j 0 ) | h ( a ) whenever uo_ < a _< a Q f o r i = l,2,...,m f o r which i ^ ^ 1, k = 1,2,...,m and and a l l s e q u e n c e ( l ] _ * l m ) n b ^ l J Jn-m^ hw < C | o b ( i 0 , J ) | h(b) w h e n e v e r b Q _< b < o i + f o r j = m + l , . . . , n and a l l s e q u e n c e s U i ' J n - m ) s u c h t h a t J k ^  3> k = 1>2>--->n- m. Conditions (2.11) - (2.17) are a c t u a l l y s u f f i c i e n t t o obta i n convergence of the eigenvalues of (1.3) to those of (1 .2), 12. However, the stronger assumptions (2.7) - (2.12) w i l l he needed to obtain the uniform estimates of section 3. We have the f o l -lowing theorem: Theorem 1. Let uu and uu be s i n g u l a r i t i e s of L as described l n section 1. Let X be an eigenvalue of (1.2) possessing k ortho-normal eigenfunctions. Then under the assumptions (2.7) - (2.12) (or under the weaker conditions (2.11)-(2.17)) there exists a rectangle R Q and a constant C on R Q such that at least k perturbed eigenvalues of (1.3) s a t i s f y l » 4 - X ' <• C ^ a + V whenever [a,b] e R0. Proof: Let G a b ( s , t ) be the Green's function for the operator k« L Q associated with the boundary conditions of (0.6) and l e t G ^ be the l i n e a r transformation on H[a,b] defined by ,b Gaby - G a b ( s , t ) y ( t ) k ( t ) dt, y e H[a,b] It is'well-known ([3], p. 1 9 2 ) that for any function y e H[a,b], the function w = G &^y i s the unique solution i n D[a,b] of the d i f f e r e n t i a l equation LQw = y. For X an eigenvalue and x any corresponding normalized eigenfunction of ( 1 . 2 ) , define a function f on [a,b] by ( 2 . 1 9 ) f = x - Y G a b* where y = X - lQ . It i s e a s i l y v e r i f i e d because of the l i n e a r i t y of a l l the operators involved that f i s a solution of the boundary value 1 3 . p r o b l e m ( 2 . 2 0 ) L Q f = 0 , U^f = U^x , i = 1,2,...,m, u j f = U^x , 1 = 1 , 2 , . . . ,n-m . We c a n f i n d t h e s o l u t i o n f c f ( 2 . 2 0 ) i n terms o f t h e b a s i c s o l u -t i o n s i n t h e f o l l o w i n g way: We a p p l y t h e b o u n d a r y c o n d i t i o n s o f ( 2 . 2 0 ) t o t h e e q u a t i o n f - I A k *k k = l t o o b t a i n the non-homogeneous s y s t e m of" l i n e a r e q u a t i o n s n ( Uax -I A k U k - 1 = 1 - 2 , . . . , m k = l n ^ - u b x - = I A k u b > k * - i = i - 2 ^ - - » n - m k = l and a p p l y Cramer's r u l e . I f f o r e a c h k, k = 1 , 2 ,...,n, t h e d e t e r m i n a n t c o r r e s p o n d i n g t o i s exp-'ftnded b y t h e c o m p l e m e n t a r y m i n o r s h ( ) h b ( ), 6 & ( ) , 6 f e ( ) (See [ l ] , c h a p t e r 4 ) , t h e n ( e x c e p t f o r t h e + s i g n s as i n d i c a t e d ) f has a r e p r e s e n t a t i o n o f t h e f o r m ( 2 . 2 1 ) f ( s ) = K(a,b) 1(a) • U a U>) b ' ' + s V ^ n - m - l ) / } ( c ) '^b 1(d) n a k J w h e r e : ( i ) ' K(a,b) d e t A(a,b) , V J l J t ] n - m ) 14. where S i n d i c a t e s summation over a l l p o s s i b l e d i s j o i n t sequences ^ l ^ m ) a n d (h'^n-m^ s u c h t h a t ^ l ^ m ^ ^ C 1*™)? ( i i ) E i n d i c a t e s summation over a l l p o s s i b l e sequences ( i , , i (a) ( i i i ) E i n d i c a t e s summation over a l l k , k = l , 2 , . . . , n and a l l p o s s i b l e sequences ( J ] _ * J n m ) such t h a t under the summation E , k 4 i ^ j „ f o r r = 1 , 2 , . . . , m - l , s = 1 , 2 , . . . , n - m ; (a) r s ( i v ) E i n d i c a t e s summation over a l l p o s s i b l e sequences (Jv<J n . ( c ) (v) E i n d i c a t e s summation over a l l k , k = l , 2 , . . . , n and a l l W p o s s i b l e sequences ( i ^ ^ ) such t h a t under the summation I , k ft 1 ^ j f o r r = 1 , 2 , . . . , m , s = 1 , 2 , . . . , n - m - 1 . (c) r 5 The s o l u t i o n f of (2 .20) as g i v e n by ( 2 . 1 9 ) or (2 .21) i s u n i q u e . In f a c t , i f g i s any s o l u t i o n of ( 2 . 2 0 ) , the f u n c t i o n h = g - f s a t i s f i e s L h » 0 , U & h » 0, i = . l , 2 , . . . , m , u j h = 0, i = 1 , 2 , . . . , n - m . S i n c e i,Q i s n o n - r e a l and hence not an e i g e n -v a l u e of ( 1 . 3 ) , i t f o l l o w s t h a t h must be the z e r o f u n c t i o n or g = f . U s i n g (2 .14) and ( 2 . 1 5 ) we can f i n d numbers a Q , b 0 (which we'may suppose c o i n c i d e w i t h the o r i g i n a l c h o i c e s of a Q , b o ) such t h a t K ( a , b ) i s bounded away f r o m z e r o on < a i . a o ' b o ~ b < ®+ ' N o w u s i n S ( 2 -^) and ( 2 . 5 ) i n (2 .21) and the f a c t t h a t K ( a , b ) i s bounded away f r o m z e r o , we can f i n d a c o n s t a n t C such t h a t f o r uu_ < a _< a Q , b Q _< b < u)+ , 15. (2.22) ||f||* < C ^ ^ ( V V l ) ^ VJ l ' J n - m ) + S ( 2 ) -©b («J1>«Jn..m) ft + ( 3 ) 1 b ( j l , J n - » - l ) — + S "^a ^ W n7 where: ( i ) T. indicates summation over a l l i , i = 1,2,...,m and a l l (1) possible sequences ( i ^ ^ ^ a n d (31>3nm.m)  s u c n t h a t i £ J' ^ i for r = 1,2, ...,m-l , s = 1,2,...,n-m; r s ( i i ) £ indicates summation over a l l j 3 j = m+1,.,.,11 and a l l (2) possible sequences ( i ^ , i m _ ^ a n d ( j ' i ^ J n _ m ) such that i„ £ J' ^ j for r = 1,2,...,m-1, s = 1,2,...,n-m; ( i i i ) £ indicates summation over a l l J, J = m+l,...,n and a l l (3) possible sequences ( i 1 , i m ) and (Jj*J n_ m -.]_) such that i„ ^ j«, ^ J f o r r = l,2,...,m , s = 1,2,. . . ,n-m-l; (Iv) E indicates summation over a l l i , i = 1,2,...,m and a l l possible sequences ( 1 ^ ? ^ ) and (J^'^n-m-l^ s u c h t n a t i ^ j _ ^  i for r = 1,2,...,m , s = 1,2,...,n-m-l. i s Clearly from (2.11) and (2.15) each term i n the sum £ of (2.22) (1) w i l l be such that ^ ( W l 1 0[-©a(2,m)j 16. as a,b -* except when 1=1, ( i - ^ i ^ )=(2,m) and ( j - i j j ' m) = (m+l,n) In which case the term i s exactly-0^(2,m). JL IT ""in ci Hence we have ^ a ^ 1 ! ' m-l ; h7 (2 .23) (1) = 0[^(2,m)] as a,b -* «)_,tt) . We now show that the second sum, £ , i n (2.22) s a t i s f i e s (2). 1 ) 6 a ( 1 l , V l ) * V J l , J n - m ; h (2.24) E (2) a o[^ a(2,m)] as a,b iu_,ua . For a given sequence (i]_> i m _ ] _ ) s e l e c t an i n t e g e r i , 1 _< i _< m, such that i £ i 1 , . . . , i m _ 1 . Then by (2.4) and (2.11) a as a u)_. Since 4 H(t«_,cJ, i t f o l l o w s that 6 a ^ 1 l , 1 m - l ^ r.Xr^fO - = o[^ a(2,m)] as a -* t»_. Using t h i s r e s u l t and (2.17)_, (2.24) now f o l l o w s . The same reasoning used i n e s t a b l i s h i n g (2 .23) and (2.24) can be used t o obta i n (2 .25) and (2 .26) r (3) CO h ( i , , i ) a v 1* m' a v 1* m; TV. = 0[-e£(m+l,n-l)] = o[-©P(m+l,n-l)] as a,b -» ou_,m+ 1 7 . We now apply (2.6), (2.23)-(2.26) to (2.22) to obtain the estimate (2.27) H f | l a < C{-Q& + 49 f e } for uo_ < a < a Q , b Q _< b < a> It follows from (2.19) and (2.27) that for any eigenfunction x associated with the eigenvalue X, (2.28) ||x - YGabx||^ < C ( ^ +^b)||x||a whenever [a,b] e R . o Let u J = p. a b denote the J-th eigenvalue (counting 1 2 m u l t i p l i c i t i e s ) of the regular problem (l .3)>|u |j<|u | _< ... and l e t y . denote the corresponding eigenfunction, chosen so that J {y .} i s an orthonormal basis i n H[a,b], J Lemma 1. Let P ( 6 ) be the projection mapping from the Hilbert space H[a,b] onto i t s subspace H f i[a,b] ( 6 > 0) spanned by a l l the eigenfunctions y. of (1.3) such that the corresponding J eigenvalues uJ' s a t i s f y ju^ - x| _< 6 . Then for any w e H[a,b], ||w - P ( 6)w|l a < (1 +-4fl)||w - YGabw||a . The proof i s given i n ([11], p. 1554). From (2.28) and lemma 1, we see that there exists a constant C on R Q such that (2.29) IIx - P ( 6)x|' a < | ^ ( - 9 A +-Ob)!lx||^ . With the choice 6 = 6 , = C(-Q + ) we obtain (2.30) Dx - P ( 6 )x| l a < |||x||a 1 8 . and hence P(&)x = 0 I m p l i e s x = 6 on [ a , b ] . S i n c e the e i g e n -space c o r r e s p o n d i n g t o the e i g e n v a l u e X of the b a s i c p r o b l e m has d i m e n s i o n k , i t f o l l o w s i m m e d i a t e l y t h a t dim P ( 6 ) A ^ _> k. T h i s i m p l i e s t h a t t h e r e e x i s t at l e a s t k e i g e n v a l u e s ( c o u n t i n g m u l t i p l i c i t i e s ) of the p e r t u r b e d p r o b l e m ( 1 . 3 ) such t h a t | u a b - X | _< 6 a f e h o l d s whenever [ a , b ] € R . T h i s p r o v e s the theorem. Theorem 1 shows, i n p a r t i c u l a r , t h a t i f X i s an e i g e n v a l u e of the b a s i c p r o b l e m of m u l t i p l i c i t y k , then ( u s i n g ( 2 . 1 1 ) and ( 2 . 1 2 ) ) t h e r e are at l e a s t k e i g e n v a l u e s yx^ of the p e r t u r b e d 1 p r o b l e m such t h a t yx^ -• X as a ,b -* ou_,w + . i n a d d i t i o n , the e s t i m a t e of the theorem i s v a l i d u n i f o r m l y on ti)_ < a _< a Q , b ^ < b < tu , . o — + The q u e s t i o n " t h a t comes up next i s under what c o n d i t i o n s does theorem 1 y i e l d e x a c t l y k p e r t u r b e d e i g e n v a l u e s yx"^ near the b a s i c e i g e n v a l u e X when a i s near uo_ and b near cu ? T h i s r e s u l t i s o b t a i n e d i f we r e q u i r e t h a t the a b s o l u t e v a l u e o f the i - t h e i g e n v a l u e of p r o b l e m ( 1 . 2 ) i s not l a r g e r t h a n the a b s o l u t e v a l u e of the i - t h e i g e n v a l u e of p r o b l e m ( 1 . 3 ) . T h i s p r o p e r t y w i l l be r e f e r r e d t o as the m o n o t o n i c i t y p r o p e r t y . Theorem 2 . I f i n a d d i t i o n t o the hypotheses of theorem 1 , the m o n o t o n i c i t y p r o p e r t y h o l d s , t h e n f o r the b a s i c e i g e n v a l u e X of m u l t i p l i c i t y k, t h e r e e x i s t s a r e c t a n g l e R Q and a c o n s t a n t C on R such t h a t e x a c t l y k p e r t u r b e d e i g e n v a l u e s (j^L s a t i s f y 19. whenever [a,bj e R Q . Proof. Suppose f i r s t that X i s the smallest non-negative "basic eigenvalue and has m u l t i p l i c i t y k. Then since 8 ^ -• 0 as a,b, -• « 3 _ , O D + by (2.11), (2.12) and theorem 1, we can f i n d points a .b^ such that 6 , i s le s s than the minimum of a l l the differen-ce o ab ces IX1 - XJ|, X 1, \ 3 d i s t i n c t ( i , j = 1 , 2 , . . . ) , whenever (»_ < a _< a Q, b Q _< b < tu+. By theorem 1 and the monotonicity property of (1.2), at least k perturbed eigenvalues u a b l i e i n the i n t e r v a l [X, X + &aV*' Sin c e ' n a b >.  x^ f o r e a c h 3 (including m u l t i p l i c i t i e s of both the and x J ) there are at most k eigenvalues yx^ on [X, X + 6 a b l and hence exactly k. A similar statement applies to the case that X i s negative. Let kj denote the m u l t i p l i c i t y of the j - t h d i s t i n c t basic eigenvalue X^ . In order to prove by induction that there are exactly kj perturbed eigenvalues yx^ which s a t i s f y l^ab " X I < 6ab (J - I- 2,...) assume that t h i s i s true for each integer j _< n. In the case n+1, 3 = 1 n+1 n4-2 n+1 that JX | < |X | there are at most > ; k. eigenvalues ^ab w n l c n s a t i s f y It then follows from the induction assumption that there are at most k n + ^ eigenvalues yx^ s a t i s f y i n g 20. and hence e x a c t l y k n + ^ by theorem 1. In the other case \ n + 2 = - \ n + 1 , there a r e , k n + ^ + ^-n+2 eigenvalues |i ^  s a t i s f y i n g l x n + 1 ! < l , a b l < I X n + 1 | + 6 a , and again by theorem 1 there are e x a c t l y k eigenvalues (j ^  near X1"1""1" and k n + 2 eigenvalues |i ^ near X n + 2 . This completes the proof of theorem 2. Theorem 3. Let the hypotheses of theorem 2 be s a t i s f i e d . Then corresponding to the eigenvalues X and u & b of theorem 2, there are orthonormal e i g e n f u n c t i o n s x^ on [a,b] a s s o c i a t e d w i t h X and y*' as s o c i a t e d w i t h the such t h a t " yab ' x J | l a ± C ( - G a + " V ' J = 1 , 2 , . . . ,k, whenever [a,b] e R . o Proof: Let {y^} be a set of orthonormal eigenfunctions on [a,b] corresponding t o the set of eigenvalues { u a b l i n theorem 2. Then Hg[a,b] i s k-dimensional by theorem 2 and P ( 6 ) x = 0 i m p l i e s x = 0 by ( 2 . 3 0 ) . Hence there e x i s t s k unique l i n e a r l y inde-pendent eige n f u n c t i o n s corresponding t o x which P ( 6 ) maps i n t o the orthonormal e i g e n f u n c t i o n s y J and by (2 .29) z' Since 2 1 . by the Schwarz I n e q u a l i t y , f o r i , j = l , 2 , . . . , k where 6 ^ denotes the Kronecker d e l t a . S i n c e the 7? are l i n e a r l y i n d e p e n d e n t , an o r t h o n o r m a l sequence can be c o n s t r u c t e d by the Schmidt p r o c e s s as l i n e a r combina-t i o n s of the 7? and i t i s e a s i l y v e r i f i e d t h a t 11 x J - z J l l a = 0 ( ^ a +-Qb). Hence |JxJ - y J || a = o(-e a+-e b) , j = i , 2 , . . . , k . 3 . U n i f o r m e s t i m a t e s f o r e i g e n f u n c t i o n s on [ a , b j . The p u r p o s e of t h i s s e c t i o n i s to o b t a i n u n i f o r m e s t i m a t e s on [ a , b j f o r the e i g e n f u n c t i o n s of theorem 3 when u)_ and u>+ are s i n g u l a r i t i e s f o r L as d e s c r i b e d i n s e c t i o n 1 . F o r the second o r d e r case when tu_.>tu+ are b o t h l i m i t p o i n t s i n g u l a r i t i e s ( a c c o r d i n g t o W e y l ' s c l a s s i f i c a t i o n ( [ 3 ] , p . 2 2 5 ) ) such u n i f o r m e s t i m a t e s are g i v e n i n [ 1 3 ] - The c e n t r a l c o n d i t i o n s used are such t h a t the p o s i t i v e f u n c t i o n g a b ( s ) d e f i n e d by ( 3 - D g a b ( s ) = f | G a b ( s , t ) | 2 k ( t ) d t a i s u n i f o r m l y bounded on a < s < b p r o v i d e d m < a < a Q , b „ < b < ID, . o — + F o r the n - t h o r d e r case we p r o c e e d i n a s i m i l a r manner. 22. F i r s t s u f f i c i e n t c o n d i t i o n s w i l l be s t a t e d f o r the u n i f o r m boundedness of (3.1) on a _< s _< b when a_< a Q , b Q _< b . These w i l l be a p p l i e d t o a s u i t a b l e f o r m of the G r e e n ' s f u n c t i o n , G a f e ( s , t ) , w h i c h w i l l a l s o be d e v e l o p e d i n t h i s s e c t i o n . The boundedness of (3.1) w i l l be o b t a i n e d as a r e s u l t and f i n a l l y the d e s i r e d u n i f o r m e s t i m a t e s . L e t W(s) be the " W r o n s k i a n " d e t e r m i n a n t , i . e . , ¥ ( s ) = det ( c p ^ " 1 ^ ( s ) ) , i , J « l , 2 , . . . , n and l e t C ± ( s ) denote the ( s i g n e d ) c o f a c t o r of c p j n _ 1 ^ ( s ) i n W(s) . L e t * C . ( a ) * C , ( b ) (3.2) a a ( i , j ) = ^ T a 7 a n d a b ( l , j ) - ^ y . F o r (i-^.*im_]_) any sequence and p any i n t e g e r we l e t ^ ( i - p i ^ - ^ p ) i denote the d e t e r m i n a n t of the m a t r i x w h i c h has Urn- i n the i - t h a x k row and k - t h column, I = l , 2 , . . . , m , k = 1 , 2 , . . . , m - 1 and U.cp cl P I n the i - t h row and m-th column. S i m i l a r l y f o r ( , J n - m _ ^ ) any sequence we l e t ^ ( j ^ - J ^ , m - l , c ^ denote the d e t e r m i n a n t of the m a t r i x w h i c h has ufcp . i n the i - t h row and k - t h column, k i s= 1 , 2 , . . . , n -m, k = 1,2, . . . , n-m -1 , and U^ip^ i n the i - t h row and n - m - t h column. A s s u m p t i o n s : In a d d i t i o n t o (2.7) - (2.12) we make the f o l -l o w i n g a s s u m p t i o n s : ( i ) u) ,ao + are not a c c u m u l a t i o n p o i n t s of the z e r o s f o r . C ^ ( s ) , I = l , 2 , . . . , n , and f o r I = 1 , 2 , . . . , m , j = m + l , . . . , n (3-3) h ( a ) | a _ ( i , j ) | , h ( a ) | a ( i , j ) | are n o n - d e c r e a s i n g on uu_' < a _< a Q , and 23. ( 3 . 4 ) h ( b ) | a b ( i , j ) I , h ( b )\o^{l, j ) | are n o n - i n c r e a s i n g on b Q _< b < uu+ . ( i i ) The f o l l o w i n g f u n c t i o n s are bounded on uT < s < ~ ( 3 . 5 ) h ( s ) T i ( s ) h(s) qp±(s) h ( t ) C.(t) P Q ( t ) W(t) h ( t ) C\j(t) P 0 ( t ) W(t) » i * = 1 , 2 , . . . , m ; , i , j = m+1, . . . , n . ( i i i ) There e x i s t i n t e g e r s i ^ , where 1 _< i ^ <_ m, m+1 _< <_ n such t h a t ( 3 . 6 ) h(aj r t a C 1 T ( a - J i s bounded on eo < a _< a Q f o r j = m+1,. . . , n and a l l sequences ( ^ i * i m _ i ) * a n d ( 3 . 7 ) h ( b ) ru c . , ( b ) i s bounded on b Q < b < uu+ f o r i = 1 , 2 , . . sequences U ^ i ^ ^ j . ) ' ,m and a l l S i n c e t h e r e e x i s t s a p o s i t i v e c o n s t a n t c such t h a t h ( s ) _> c, oo < s < «o+ , we have the f o l l o w i n g o b v i o u s i n e q u a l i t i e s on uo_ < s < m ; ( 3 . 8 ) |cp i (s)| _< C h ( s ) |cp i(s) | , i = l , . . . , n : 24. ( 3 . 9 ) and C . ( t ) p c ( t ) w(t ) c . ( t ) * s p Q ( t ) W(t) < c < c h ( t ) c±(t) p Q ( t ) w(t) h ( t ) Cj (1>) p Q ( t ) w(t ) i = 1 , . . . , m ; j = m+1,...,n We now p r o v e some fundamenta l i n e q u a l i t i e s w h i c h w i l l he needed t o o b t a i n the boundedness of ( 3 . 1 ) . Lemma 2. F o r k « l , 2 , . . . , n l e t ( 3 . 1 0 ) rPk(s) = C k ( B ) = |op k(s)| i f OD_ < s < a Q , b Q < s < m+ if a 0 < s < V C k ( s ) | i f (D_ < s < a Q , h D < s < UJ + i f a „ < s < b n o — — o Then t h e r e e x i s t s a c o n s t a n t C , independent of a , b as w e l l as s, such t h a t ( 3 . 1 1 ) TT <Pi(s) < C • ^ , (s ) h ( s ) d o on a < 3 _< b , a _< a Q f o r a l l i n t e g e r s i, 1 _< i _< m and a l l sequences ( i ^ , i m ) such t h a t i ^ i ^ , , k = l , 2 3 . , . , m ; ( 3 . 1 2 ) ' W W q»j(s) _< C • y ± (s ) h ( s ) o on a < s < b , b Q _< b f o r a l l i n t e g e r s j , m+1 _< j _< n and a l l sequences (J]_*J n_m) such t h a t j ^ J k , k = 1 , 2 , . . . , n - m ; 2 5 . ( 3 . 1 3 ) n. Cj(.) < C • ^ ( s ) h ( s ) on a < s < b j a .1 a 0 f o r a 1 1 i n t e g e r s j , m+1 _< j _< n and a l l sequences ( i j - i j n . i ) ? ( 3 . 1 4 ) V^n-m-l'1) C ± ( s ) < C • £ , , ( s ) h ( s ) on a < s < b , b Q _< b f o r a l l i n t e g e r s i , 1 _< i _< m and a l l sequences (J^J^-^^)-P r o o f . We f i r s t note t h a t by ( 3 - 3 ) and ( 3 . 4 ) ( 3 - 1 5 ) | a a ( i , j ) | h ( a ) < | a s ( i , j ) | h ( s ) , | a * ( I , J ) | h ( a ) < | a * ( i , j ) | h ( s ) when t»_ < a _< s _< a Q , and j a b ( i , j ) | h ( b ) < | a s ( i , j ) ' j h ( s ) , | a £ ( i , j ) | h ( b ) < | a | ( i , J ') j h ( s ) when b Q <_ s < b < ou+ f o r i = 1 , 2 , . . . ,m , j = m+1,. . . , n . From the f a c t t h a t h ( s ) _> c f o r some p o s i t i v e c o n s t a n t c , i n e q u a l i t i e s ( 3 . 1 1 ) - ( 3 - 1 4 ) are c l e a r l y v a l i d on [ a , b ]. To p r o v e ( 3 . 1 1 ) on tu_ < a _< s _< a Q we have by (2 .18) and ( 3 - 1 5 ) t h a t h. <p±(s) _< C | a a ( i , j 0 ) < p 1 ( s ) | h ( a ) < C | a s ( i , J 0 ) « p 1 ( s ) j h ( 8 ) = c U . . ( s ) | h ( s ) . J o Thus ( 3 . 1 1 ) h o l d s on uj_ < a _< s _< a Q as w e l l . F i n a l l y ( 2 . 1 3 ) ad ( 3 . 4 ) y i e l d | a g ( i , j j | h ( s ) < C on b Q < s < hence ( 3 . 1 1 ) i s v a l i d on [ b 0 , t u + ) . Thus ( 3 . 1 1 ) i s v a l i d on the whole i n t e r v a l a j s S < u o + , a _ < a Q . The p r o o f of ( 3 . 1 2 ) i s completed i n the same way and w i l l be o m i t t e d . 26. To prove ( 3 . 1 3 ) on < a j< s jC a we deduce from ( 3 . 6 ) that a < C|a»(lJ,3)|h(a) and hence by ( 3 . 1 5 ) that W W * * ) c j ( s ) i C l 0 ^ 1 © ^ ) Cj ( 8 ) | h ( a ) - 0 1 ^ , ( 8 ) ^ ( 8 ) , Hence ( 3 . 1 3 ) i s v a l i d on ^ < a < B < a D. F i n a l l y ( 3 . 1 3 ) holds f o r b 0 < s < » + as weU since | a * ( i ^ j ) j < C on b^< s < o>+ by ( 3 . 4 ) . Thus ( 3 . 1 3 ) i s v a l i d on the whole i n t e r v a l , a 1 S < a _< a Q. The proof of (3.1*0 i s completed i n the same way and w i l l be omitted. We now construct the Green's function f o r k«L Q associated with the boundary conditions of ( 0 . 6 ) . The method used w i l l be the one given In [3l> page 192. Let 0(a,b) denote the deter-minant of A(a,b) and n &(a,b) the determinant of the matrix obtained from A(a,b) by replacing the j - t h column by £ C ± ( t ) U l ^ ? C i ( t ) ^ Z e r ° S ) I p r t(t) W(t) L p f t ) W(-t) » °» 0 • 1=1 ° i = l ° S i m i l a r l y l e t flj|(a,b) be the determinant o f the matrix obtained from A(a,b) by rep l a c i n g the J-th column by (m zeros) _ , _ „ „ , ' , * C ± ( t ) J%9± " C ± ( t ) D g " ^ o , o , . . . , o , l P o ( t ) W ( t ) * L P n ( t ) W(t) i = i i»i 2 7 . r n We set (t ^ W t t ) I M 8 ) C ± ^ ) ' a < t < s < b K a b ( s , t ) = \ i= l , a < s < t < b . and determine constants Ay 3 = l , 2 , . . . , n such that the function n ( 3 . 1 6 ) s a t i s f i e s the boundary conditions of ( 0 . 6 ) (as a function of s), The r e s u l t i n g function G a b ( s , t ) Is then the required Green's function. Lemma 3 . The Green's function G a b ( s , t ) f o r fc> L Q associated with the boundary conditions of (0.6) can be expressed i n the n firbrjl °a>' b> * j ( s ) > a < t < s <b form: r ( 3.17) G a b ( s , t ) n -nTi7b7l ab( a^) «Pj( B)' a < s < t < b J=i Proof. Applying the boundary conditions of ( 0 . 6 ) to ( 3 . 1 6 ) , we obtain the following non-homogeneous system of n l i n e a r equations: n i = 1 , 2 ,...,m 1 A j ¥ j = "2, p n ( t ) W(t) ' i - 1,2,...,n-m. k=l Then Cramer's rule y i e l d s A, = - f i b ( a ^ b ) , j « 1 , 2 , . . . , n n ( a , b ) 2 8 . Thus for s < t (for which K a b ( s , t ) = 0) i t follows Immediately that n (3.18) G a b ( a , - t ) - . — i - ^ ^ b ) V j ( B ) . For t <: s, Oab* 8'* 5 = W8^ - T ^ ) I ° b ( a ' b ) * J < s ) ' J - l Subtracting and adding n • J - l to the r i g h t side we obtain n (3.19) Q a b ( B , t ) = K a b ( s , t ) - T T r | 7 F 7 5 ; ( f i J ( a , b ) + O ^ b ) ) ^ ) J - l n + QTaTE) I n a < a > b ) <Pj(8>-J - l Consider the second term on the r i g h t of (3.19); By elementary operations on determinants one may show that i / 1 -j \ v C,(t)cp,(s) TSfk^) (nj|(a,b) + O a(a,b); VjCa) = p ^ t ) w(t) . Hence summing over j (as i n (3.19)) we obtain that n - p n ( t ) V ) ' I *j( 8> - K a b ( a,t) . ° J - l 29. Using this result in (3 . 19)* it is clear that n G ^ 8 ^ - TT[alb)I Q i ( * > h ) * j < s ) ' t ^ s j - 1 This and (3.18) prove- the lemma. If the determinants in (3-17) are expanded by complementary minors (see [ l j , chapter 4]) using the Ti ( ), rvb( ) notation then it can be shown that G a b( s*t) has the following form: For a £ t <, s _< b, ( 3 . 2 0 ) G a b(s,t) = 1 V ft (+) n a ( W r q ) C Q ( t ) Vvm W J n-m> , X&*jl \L (±} na p0(t)\t) * J[L(±) V where: (i) Z indicates summation over all possible sequences (ii'Vi> : (VW; (ii) Vindicates summation over all q, q = l , 2 , . . . , n such q that q ^  i k , k = 1 , 2 , . . . ,m-l; ( i i i) indicates summation over all p, p = l , 2 , . . . , n and all possible sequences (Ji*Jn_m) such that p ^  j ^ i for r = 1 , 2 , . . . , n - m , k = 1 , 2 , . . . , m - 1 ; (iv) The signs In front of h-a( ) and hb( ) may be deter-mined by the relative position of the corresponding minors in ( 3 . 1 7 ) and also by the arrangement of the columns in 3 0 . these minors. For our purposes i t i s unimportant to know which sign to use (as w i l l he seen l a t e r ) . For a £ s < t _< b, ( 3 . 2 1 ) G a b ( B , t j -^a7B7 . i U. (±) h b p Q ( t ) W(t) /(£ ( ± } — h ~ V8'/ ( Jl'4i-m-l) q where: ( i ) YJ indicates summation of a l l possible sequences ( J l , J n - m - l ) ( J'l , Jn-m-l^ ; i ( i i ) ^ indicates summation over a l l q, q = l , 2 , . . . , n such q. tha,tq ^ j k , k » 1 , 2 , . . . , n - m - l ; ( i i i ) ^ indicates summation over a l l p, p = l , 2 , . . . , n and a l l possible sequences ( i ] _ * i m ) such that p ^ i r ^  J k for r = l , 2 , . . . , m , k = 1 , 2 , . . . , n - m - l ; ( i v ) The signs i n front of n & ( ) and h b ( ) may be determined as i n ( 3 . 2 0 ) but are unimportant for our purposes. Proof of, 5 . 2 0 ; For any sequence {^'^m-l^ l e t ^ a ^ l ^ m - l ^ d e n o t e the following determinant: 31. n I c £ t ) u V P 0 ( t ; w(t) a*i- ^ 1 m-1 n I C (t) T T 2 p n ( t ; w(tj U a * i 1 m-1 n I P 0 ( t ) W ( t ) m-1 Then i f each determinant Q!:(a,b), j = l , 2,...,n i s expanded a. by complementary.minors i t i s clear that the sum (3. 22) n £ O a(a,b) «pj(s) j=l w i l l consist only of terms of the form h*( ) ) cpk(s). Let ( i j j i ^ i ) be any fixe d sequence. Then except for interchange of columns, the cofactor h*(i-,,i„, •,) w i l l occur i n exactly a^ 1 m-l' n - m + I determinants n^(a,b). In f a c t the cofactor h * ( i , i .. ) w i l l occur In exactly those determinants 0^(a,b) for which a j ^ i , , k = 1 , 2 , . . . , m-1. Factoring t h i s cofactor h * ( i , , i , ) jv a' x ni"" x out of the determinants Q^(a,b), j ^  i , , k « 1 ,2, . . . ,m-1, i n .a ic the sum (3.22) we obtain the product (?• 2?) { I (±) W W <Pp< 8 ) } ^ W l ' where the summation ^ Is described by ( i i i ) under (3.20). * Note that for each p ^  i k there i s exactly one sequence ( O i * J n _ m ) 32. s a t i s f y i n g these conditions. We now expand the cofactor h * ( i n , i .. ) to obtain a x l m-l (3.24) h S ( i 1 , i m . 1 ) =1 (+) n ^ i ^ . p q ) C f f H ( t ) q where the summation^ i s described by ( i i ) under (3.26). Prom (3.23) and (3.24) i t now follows that the term i n (3.22) corresponding to the cofactor ^ a ^ l ' ^ n i - l ^ i s ( l (+) Vtl'Vl'-J) w(tj)(Z <±)Vl'W Vs)) • Since every term i n the expansion of (3.22) involves a cofactor h*( ), we obtain for t < s that where the summation ^ i s described by ( i ) under (3.20). To complete the proof of (3.20) we divide numerator and denomina-tor of the r i g h t side of (3.25) by h-a'hb. A similar proof holds fo r (3.21) and w i l l be omitted. Lemma 4. Assume that conditions (2.7) - (2.10) and (3.3) - (3.7) are s a t i s f i e d . Then the p o s i t i v e function S a l o(s) defined by 33. 4 < s > - J . i < w..t ) i 2 k(t)« i s u n i f o r m l y bounded on a _< s _< b p r o v i d e d a _< a Q , b Q _< b. Proof. We f i r s t express g & b ( s ) i n the form g a b ( s | = ^ j G a b ( s , t ) | 2 k ( t ) d t + | V a b ( s , t ) i 2 k ( t ) d t j *2. from which f o l l o w s t h a t (3.26) g a b ( s ) < ( j V a b( S,t)| 2k(t)at^ + ( j j G a b ( S ) t ) | 2 k ( t ) d ^ U s i n g ( 3 . 2 0 ) and the Minkowski i n e q u a l i t y we o b t a i n t h a t 1 ( 3 . 2 7 ) ( |Gab(s,t)rk(t)dt)^  < |K(a,b) j ^ ( ± ) n a P 0 ( t ) W ( t ) By ( 3 . 1 2 ) each term i n the sum £ i n v o l v i n g cp ( s ) , p > m, can be r e p l a c e d by a term i n v o l v i n g cp. ( s ) h ( s ) and by ( 3 . 1 3 ) each o term i n T, I n v o l v i n g C ( t ) , q > m, can be converted i n t o a term q " i n v o l v i n g C . t ( t ) h ( t ) . Having done t h i s , we o b t a i n , u s i n g (3.8) o and ( 3 . 9 ) t h a t t h e r e e x i s t s a constant C such t h a t ( jV r t(«.t)l 8k(t)«)* < m |K(a,b)| £ 3 m * I l Y i d | h ( s ) ^ . ( s ) C j ( t ) h ( t ) P 0 ( t ) W(t) a where uu_ < a _< a Q , b Q _< b < ou+ and Y J J are bounded f u n c t i o n s of a and b on oo_ < a _< a Q , b Q _< b < OJ +. Prom ( 3 . 5 ) and the f a c t t h a t K(a,b) i s bounded away from 0 on UJ_ < a < a Q , b Q _< b < i u + 3 4 . f o l l o w s t h a t ( f V a b ( B , t ) | 2 k ( t ) d t ^ 1 "2" I s u n i f o r m l y bounded on a < s < a _< a Q , b Q <_ b . The p r o o f t h a t the second term on the r i g h t of ( 3 . 2 6 ) i s u n i f o r m l y bounded on a _< s _< b , a _< a Q , b Q _< b i s s i m i l a r and w i l l be o m i t t e d . T h i s g i v e s the d e s i r e d r e s u l t . Theorem 4 . Assume t h a t are s i n g u l a r i t i e s f o r L as d e s c r i b e d i n s e c t i o n 1 and t h a t assumptions ( 2 . 7 ) - ( 2 . 1 2 ) , ( 3 - 3 ) - ( 3 - 7 ) are s a t i s f i e d . I f i n a d d i t i o n the m o n o t o n i c i t y p r o p e r t y i s s a t i s f i e d , t h e n the e i g e n f u n c t i o n s x^ c o r r e s p o n d i n g t o \ and y & b c o r r e s p o n d i n g t o jj^y, i n theorem 3 are such t h a t lab ( 3 . 2 8 ) y a b = x J ( s ) - f J ( s ) + 0 ( - 9 a ) + O ( ^ ) , j = 1 , 2 , . . . , : f o r a < s < b , oo_ < a £ aQ , b 0 _< .b < uo+ where f J ( s ) i s the unique s o l u t i o n of ( 3 . 2 9 ) L f = iQf , U a f = b a x j , i = l , 2 , . . . , m , U^f = u j x j ' , i = l , 2 , . . . , n - m . P r o o f . The Schwarz I n e q u a l i t y f o r H [ a , b j y i e l d s y i b ( s > -ab y i b - x J Jllb a 35. Hence lemma 4 and theorems 2 and 3 show t h a t t h e r e e x i s t s a constant C such t h a t (3 .30) | y a b ( s ) - (X - ^ 0 ) G a b x J ( s ) | < C ( ^ a + ^ ) on a _< s < b, whenever a £ a Q , b Q _< h. The s o l u t i o n f J ( s ) of the boundary-value problem (3 .29) i s g iven by (2 .19) or ( 2 . 2 l ) w i t h x r e p l a c e d by x J . The f u n c t i o n F J d e f i n e d by F J ( s ) = (x - ^ o ) 0 a b x J ( s ) - x J ( s ) + f J ' ( s ) s a t i s f i e s L F J = - t Q F J , U ^ ' = 0, i = 1 ,2 , . . . ,m, U bF J' = 0, i = 1 , 2 , . . . ,n-m and hence F J i s the zero s o l u t i o n on a < s < b f o r j = l , 2 , . . . , k . T h i s w i t h ( 3 . 3 0 ) immediately g i v e s the uniform estimates ( 3 . 2 8 ) . 4. Asymptotic v a r i a t i o n a l formulae f o r e i g e n v a l u e s . The purpose here i s t o d e r i v e formulae f o r the change j i a b - X of e i g e n v a l u e s under the p e r t u r b a t i o n D -» D[a,bJ, v a l i d f o r a,b i n neighbourhoods of O J _ , I H + r e s p e c t i v e l y . Let x J, y J denote the normalized e i g e n f u n c t i o n s a s s o c i a t e d w i t h X and p.J = as d e s c r i b e d i n theorem 3 and l e t f J be the unique s o l u t i o n of ( 3 . 2 9 ) . We then have the f o l l o w i n g theorem: 36. Theorem 5. Under the assumptions of theorem 4 the f o l l o w i n g asymptotic v a r i a t i o n a l formulae f o r the eigenvalues X, u a V j are v a l i d : (4 .1) X - u a b = [ f V ] ( b ) - [ f J x J " ] ( a ) + ( l Q - X ) ( f J " , f 3 ) \ + (*a + ^ ) ( f J , l ) b O ( l ) as a,b -• U J _ , O O + . Proof. Let Uy = 0 denote the s e l f - a d j o i n t set of boundary co n d i t i o n s given by (0.6) and (0.7). Then by [3\> chapter 11, there e x i s t boundary forms U , U Q of rank n such that [ u v](b) - [uv](a) = Uu-U*v + U Q U . Uv f o r any p a i r u,v € C n ~ 1 [ a , b ] , where • represents the "dot" product. Now Uy J = 0 by (0.6) and (1.3) and Ux J = UfJ' by (3.29), hence (dropping the s u p e r s c r i p t s j ) [xy](b) - [xy](a) = Ux- U+y = [ f y ] ( b ) - [ f y ] ( a ) . Then a p p l i c a t i o n of Green's formula (0.5) t o the d i f f e r e n t i a l equations Lx = >,x , Lf = £ f and Ly = uy on [a,b] leads t o (4.2) (X - u ) ( x , y ) b = ( i Q - u ) ( f , y ) a ; (4.3) [ f x ] ( b ) - [ f x ] ( a ) = U 0 - X ) ( f , x ) b . We o b t a i n as a consequence of theorems 1,2 and 3 that p = X + o ( l ) and ! ( x , y ) b - (x,x) f t| < l|x||5ly-x||a = 0 ( 1 ) as a,b UJ ,uu . 37. Hence and ( 4 . 2 ) y i e l d (4 .4) X - u = U Q - X)(f,y) f t[l + o(l)]. We now appeal to the uniform estimate (3.2b) to obtain (4-5) (t,y)l = ( f , x ) i - < f , f £ + ( « a + - o b ) ( * . D * o ( i ) . Then applying ( 4 . 3 ) and ( 4 . 5 ) to ( 4 . 4 ) the r e s u l t ( 4 . 1 ) follows eas i l y . In conclusion we point out that i n many examples conditions ( 2 . 7 ) - ( 2 . 1 0 ) , (3.3) - (3.7) are s a t i s f i e d when h(s) = 1 on tu_ < s < o i + . Setting h(s) = 1 we obtain the simpler conditions ( ( 2 . 7 ) - ( 2 . 1 0 ) , (3.3) - (3.7) with h(s) = l ) which are actually s u f f i c i e n t f or the boundedness of ( 3 . 1 ) . Except for minor s i m p l i f i c a t i o n s the proof i s the same as that of lemma 4 . Also In some cases, X = 0 i s not an eigenvalue and i t i s permissible to replace iQ by 0 . Then f can be taken as a real-valued solution of L f - 0 . F i n a l l y f o r various problems the f i r s t two terms on the r i g h t of ( 4 . 1 ) dominate the other terms and the asymptotic r e l a t i o n (4.6) X - u a b ~ [ f J x J ] ( b ) - [f J x J ' j ( a ) Is v a l i d f or a,b -• uu ,uo (See chapter I I I ) , 38. 5. T h e s e c o n d o r d e r c a s e : w , w | l i m i t p o i n t s i n g u l a r i t i e s I n t h i s s e c t i o n w e s h a l l s h o w h o w o u r t h e o r y a p p l i e s t o t h e s e c o n d o r d e r d i f f e r e n t i a l o p e r a t o r L = L 2 d e f i n e d b y o n t h e o p e n i n t e r v a l U J _ < s < t u + s h e r e k , p , q a r e r e a l - v a l u e d f u n c t i o n s o n t h i s i n t e r v a l w i t h t h e p r o p e r t i e s t h a t ( i ) p i s d i f f e r e n t i a b l e ; ( i i ) k , q a r e p i e c e w i s e c o n t i n u o u s ; a n d ( i i i ) k , p a r e p o s i t i v e - v a l u e d . T h e p o i n t s uu_ a n d <u+ w i l l b e l i m i t p o i n t s i n g u l a r i t i e s f o r L a c c o r d i n g t o W e y l ' s c l a s s i f i c a t i o n ( [ 3 J , p . 2 2 5 ) . C l e a r l y t h i s o p e r a t o r i s a p a r t i c u l a r c a s e o f ( 0 . 1 ) w i t h n = 2. F o r t h i s o p e r a t o r L w e c a n a p p e a l t o a t h e o r e m o f W e y l ( [ 9 L p « 4 5 ) t o o b t a i n t h e e x i s t e n c e o f b a s i c s o l u t i o n s c p - ^ ^ o f L x = 0 s u c h t h a t o (5-1) c p 1 e H [ c , u ) + ) , c p 2 e E(w_,c] c p 1 i H (OJ_,CJ , c p 2 i H [ c , u ) + ) f o r a n y c s a t i s f y i n g uu_ < c < uo + a n d s u c h t h a t [ c p 1 c p 2 J ( s ) = 1 o n < s < uu. . — • + I t w i l l b e a s s u m e d i n t h e s e q u e l t h a t t » _ . » u ) + a r e n o t a c c u m u l a t i o n p o i n t s o f t h e z e r o s o f cp^ a n d c p 2 . L e t D 2 b e t h e s e t o f a l l x € H s u p h t h a t x e C ] ( o ) _ , u ) + ) a n d 39-x' is absolutely continuous on every closed bounded subinterval of ((M^,U)+). Then the basic problem corresponding to (1 .2) is (5.2) Lx = Xx , x e D2 . Let Dgfa^bJ be the set of a l l y e H[a,b] such that (i) y e C^fa^bJ and y' is absolutely continuous on [a,b]; (ii) Ly e H [ a , b ] j and ( i i i ) Uay = Ufey = 0 where u a y = a 0( aM a) + ai(a)y'(a) V = P 0 ( b ) y ( b ) + Pi(b)y(b) with ct^a^ real-valued functions not both zero for any a on (u> >&Q\> and with P0,P-]_ real-valued and not both zero on [bQ,u)+). Then the perturbed problem corresponding to (1.3) becomes the regular self-adjoint eigenvalue problem (5-3) Ly = uy , y e D2[a,bJ . The problem of obtaining estimates for eigenvalues and eigenfunctions of (5-3) for a,b near uo_,t«+ has already been considered by C.A. Swanson in [13J. We shall show that our assumptions used for the n-th order case specialize with l i t t l e variation to his assumptions when n = 2 (See [13], pp. 306-307). For the present case exactly one basic solution cp1 e H[C,OJ ) and the other basic solution mp e H ( U J _ , C ] , 40. ID < c < IM+, hence c o n d i t i o n ( l . l ) i s s a t i s f i e d t r i v i a l l y . The n o t a t i o n used i n s e c t i o n 2 now s p e c i a l i z e s t o the f o l l o w i n g : n a = u a c P l ; 6 a ( i ) = U a x j V i b ( i ) = U ^ . ; 6 b ( i ) = U b x . Hence assumptions (2 .7 ) - (2 .12) reduce t o the f o l l o w i n g : V v ( 5 - 5 ) TV M . - ° ( U > T T ? , M ° - o ( l ) a ' 1 D T 2 as a , b -• U J _ , U U + . A l s o (5'6) h(aj tfaVl rp2(a) ' h(b) ^(bj are bounded on neighbourhoods ID < a < a_ , b „ < b < m, of ID , uu — — o o - •+• — + r e s p e c t i v e l y . F i n a l l y i t f o l l o w s f r o m the maximum-minimum p r i n c i p l e f o r e i g e n v a l u e s [ 4 ] , [10] t h a t the m o n o t o n i c i t y p r o p e r t y f o r (5 .2 ) h o l d s . S i n c e i t i s known t h a t each e i g e n -v a l u e of (5 .2 ) has m u l t i p l i c i t y 1, theorems 1 , 2 , and 3 h o l d w i t h * a = V TT *1 M a ' * b V lcp2l C o n s i d e r i n g now the assumptions i n s e c t i o n 3 we f i r s t note t h a t (5.7) 3 1 ( s ) = -cp 2 (s ) , C 2 ( s ) = cp 1 (s) Hence assumptions ( 3 . 3 ) and ( 3 . 4 ) reduce t o 41. (5.8) h(a) ( 5 . 9 ) h ( b ) cp2(a) q^ jTa) cpx(b) < h(s) _< h( s) cp2(s) «P1(s) < a £ s _< a Q; b < s < b < it). From (5-7) i t i s clear that assumptions (3.6) and (3 .7) are a restatement of (5 .6) for the present case. F i n a l l y assumption ( 3 . 5 ) requires the boundedness of (5.10) h ( s ) cp 1 (s) ||cp 2(t) h ( t ) 11 , h ( s ) cp 2(s) | ^ ( t ) h ( t ) on t» < s < ix) + . (We are making use of the f a c t that p(s)W(s) = 1 on OJ_ < B < uu+. ) We summarize the r e s u l t s i n the following theorem: Theorem 6. Let <u_,uu+ be l i m i t point s i n g u l a r i t i e s for L and l e t \ be an eigenvalue of ( 5 . 2 ) and x the corresponding normalized eigenfunctiDn. Then, I f conditions (5-4) - (5-6) are s a t i s f i e d there exists a rectangle R Q and a constant C on R Q such that exactly one perturbed eigenvalue yx^ of ( 5 . 3 ) s a t i s f i e s Kb " x l < °(*a + V whenever [a,b] € RQ. Further, there exists a normalized eigen-function y a b corresponding to such that "y ab - X C < c ( * a + V whenever [a,b] e R . If In addition conditions (5-8) - (5-10) are s a t i s f i e d , the uniform representation y a b ( s ) = x ( s ) - f ( s ) + 0(-©J + 0(-©J, 42. i s v a l i d on a s j< b, u)_ < a _< a Q , b Q _< b < uu+ where f ( s ) i s the unique s o l u t i o n o f t h e boundary problem L c f . 0, U f tf = U ax, U b f = U bx . A l s o t h e f o l l o w i n g v a r i a t i o n a l f o r m u l a i s v a l i d as a,b -» w , u o + ; X " ^ab = t f x j ( b ) - [ f x ] ( a ) + (<t 0- X ) ( f , f ) a + (^ a + ^ b ) ( f , l ) b 0 ( l ) . P r o o f . The d i f f e r e n t p a r t s o f theorem b f o l l o w as p a r t i c u l a r c a ses o f theorems 1-5. I n many second o r d e r examples, c o n d i t i o n s (5.4), (5.6) and (5-10) h o l d w i t h h ( s ) = 1 (see [13] f o r examples). I n t h i s case assumptions (5.&) and (5.9) may be o m i t t e d ; t h e s e a r e t h e n consequences o f (5.4) and the f a c t t h a t [ c p 1 ^ 2 ] ( s ) = 1. Then t h e new c o n d i t i o n s ((5.4) - (5.6), (5.10) w i t h h ( s ) = l ) are s u f f i c i e n t f o r t h e e s t i m a t e s o f theorem 6. 6. One end p e r t u r b a t i o n problems. As a s p e c i a l case of t h e m a t e r i a l i n s e c t i o n s 1-4, we c o n s i d e r t h e o p e r a t o r L g i v e n by ( 0 . l ) on (u>_,bj, b f i x e d , b „ < b < u),. We assume t h a t t h e d i f f e r e n t i a l e q u a t i o n L^x = 0 o — + -^ o has b a s i c s o l u t i o n s cp ., j = l , 2 , . . . , n s uch t h a t ou i s not an a c c u m u l a t i o n p o i n t o f z e r o s f o r cp., j = l , 2 , . . . , n and such t h a t 43. ( 6 . 1 ) cp j $ H ( i u_,bJ , j = l,2,...,m, and cp. € H(u>_,bJ , j = m+l,...,n, l i m m Is) " = ° > i = 1,2,...,m-l. S - tt) T i V ' Let D(uj_,bJ be the set of a l l x e H(uu_,b] such that: ( i ) x e Cn"1(uu_,b] and x ^ n _ 1 ^ i s absolutely continuous on • every closed bounded subinterval of (uo_,b]; and ( i i ) x s a t i s f i e s the boundary conditions Uj^ x = 0, i = 1,2,. . . ,n-m where the boundary operators are given by ( 0 . 7 ) . Then the eigenvalue problem ( 6 . 2 ) Lx = Xx , x e D(c»_,b] w i l l be referred to as the semi-perturbed problem. As before we s h a l l suppose that ( 6 , 2 ) has at least one r e a l eigenvalue X and that lQ i s not an eigenvalue. We s h a l l (as i n the previous cases) obtain estimates for eigenvalues and eigenfunctions of (1 .3) f o r a near oa_. This w i l l be done by comparing problems (1 .3) and ( 6 . 2 ) with ( 1 .3) regarded as a perturbation of ( 6 . 2 ) . Let X be any eigenvalue and x a corresponding normalized eigenfunction of ( 6 . 2 ) . To obtain convergence of the eigen-values of ( 1 .3) to those of ( 6 . 2 ) we s h a l l require that conditions ( 2 . 7 ) , ( 2 . 9 ) and (2.11) are s a t i s f i e d . At the fi x e d endpoint b one deduces from ( 0 . 7 ) and the fa c t that tQ i s not an eigenvalue of ( 1 .3) that h V j( J ^ ,J" n_ m) w i H be constant for each sequence 44. ( j 1 , j n _ m ) and that h b(m+l,n) ^ 0. F i n a l l y the monotonicity property that w i l l be r e q u i r e d i s that the absolute value of the i - t h eigenvalue of (6.2) i s not greater than the absolute value of the i - t h eigenvalue of (1.3). We then have the f o l l o w i n g theorem: Theorem 7. Let X be an eigenvalue of (6.2) of m u l t i p l i c i t y k and assume t h a t the monotonicity property holds. Then, i f c o n d i t i o n s (2.7), (2.9) and (2.11) are s a t i s f i e d , there e x i s t s an i n t e r v a l (uo *a Q] and a constant C such that e x a c t l y k eigen-values u a of (1.3) s a t i s f y IX - n a| < C9 a whenever uu_ < a _< a Q . There e x i s t k orthonormal eigenf unctions x^ of (6.2) corresponding to \ and k orthonormal eigenfunctions y£ of (1.3) corresponding t o such t h a t a a (6.3) l|x J - y ^ l r < CO , j = l,2,...,k a" a — a whenever O J_ < a <_ a Q . Proof; Let G ( s , t ) denote the Green's f u n c t i o n f o r the operator k.L Q a s s o c i a t e d w i t h the boundary c o n d i t i o n s of (0.6) and l e t G be the l i n e a r operator on H[a,b] defined by a f,b G av(s) = I G a ( s , t ) v ( t ) k ( t ) d t , v e H[a,b]. a For x any normalized e i g e n f u n c t i o n of (6.2) corresponding t o the eigenvalue X we define a f u n c t i o n f on [a,b] by the equation f = x - vG x where Y = X - l„ . a o 45. Then LJF = 0 , U*f = U*x, i = l , 2 , . . . , m , ujf = 0 , i = 1 , 2 , . . . , n - m . In terms o f the b a s i c s o l u t i o n s f has a r e p r e s e n t a t i o n o f the f o r m : n hx <Pk(s) k=l where L i n d i c a t e s summation over a l l p o s s i b l e sequences ( i i / i m _ i ) and ( d - ^ d ^ j j s u c n t h a t i ^ j ^ k f o r p = 1 , 2 , . . . , m - 1 , q = 1 , 2 , . . . , n - m . Now h. and by (2 .14) i s c o n s t a n t f o r a l l sequences ( , J n _ m ) 7 a n d a ^ - 1 ! ' x m } = o ( l ) as a - «)_ f o r a l l sequences ( i , i ) d ( l , m ) , hence t h e r e e x i s t s a number a . o> < a^ < b v 2.: m v o - o such t h a t K ( a , b ) Is bounded away f rom 0 f o r OJ_ < a _< a Q . T h i s e n a b l e s us t o f i n d a c o n s t a n t C on (uo_,aQ] such t h a t W W n / k = l \ r T whenever o>_ < a _< a Q , where E i n d i c a t e s summation over a l l p o s s i b l e sequences ( i x j i m - l ^ s u c ^ t h a t i r ^ k , r = 1 , 2 , . . . , m - 1 . But f o r each sequence ( i - ^ , i m _ i * M , m + l j£ k _< n , t h e r e e x i s t s a sequence ( i i * i m _ i * k ' ) w i t h 1 _< k ' <: m. Then by u s i n g f|cpkHa =o^|lcpk, ll^ j a s a - OJ_ by (6 .1 ) and ( 2 . 4 ) , we o b t a i n t h a t W W h. as a - ; I H _ . T h i s i m p l i e s w i t h the h e l p of (2 .11) t h a t t h e r e e x i s t s a c o n s t a n t C and a number a Q (which may be p r e - s u p p o s e d t o be our o r i g i n a l c h o i c e ) such t h a t 4 6 . whenever «o_ < a _< a Q . The remainder of the p r o o f t h a t t h e r e e x i s t at l e a s t k e i g e n v a l u e s \i 3 of (1 .3 ) such t h a t ( 6 . 5 ) \»l - X | < G © a i s s i m i l a r t o t h a t of theorem 1 and w i l l be o m i t t e d . The p r o o f t h a t t h e r e are e x a c t l y k p e r t u r b e d e i g e n v a l u e s | i a s a t i s f y i n g (6.5) u s i n g the m o n o t o n i c i t y p r o p e r t y i s the same as t h a t of theorem 2 and a l s o w i l l be o m i t t e d . The p r o o f of the e x i s t e n c e of k o r t h o n o r m a l e i g e n f u n c t i o n s x J c o r r e s p o n d i n g t o \ and k o r t h o -normal e i g e n f u n c t i o n s y £ c o r r e s p o n d i n g t o ^ s a t i s f y i n g (6.3) i s the same as t h a t of theorem 3 except t h a t b i s f i x e d f o r the p r e s e n t case . In o r d e r t o o b t a i n u n i f o r m e s t i m a t e s of y ^ ( s ) - x J ( s ) on a < s < b , f o l l o w i n g the method of s e c t i o n 4, we need s t r o n g e r assumptions t h a n ( 2 . 7 ) , ( 2 . 9 ) and ( 2 . 1 1 ) . I t w i l l be supposed i n a d d i t i o n t h a t c o n d i t i o n s ( 3 . 3 ) and ( 3 . 6 ) are s a t i s f i e d and t h a t ( 3 . 5 ) h o l d s on u)_ < s _< b . To p r o v e the boundedness of (3 - 1 ) on ui_ < s < b by lemma 4 we need o n l y show t h a t i n e q u a l i t i e s (3 . 1 1 ) and (3 - 1 2 ) are v a l i d on a < s < b , a _ £ a 0 a n d t h a t i n e q u a l i t i e s (3 . 1 2 ) and (3 . 1 4 ) are v a l i d on a < s < b , to < a < b . L e t (6.6) $j(3) = I f a < s < a^ — o i f a Q _< s _< b r | C . (s ) | i f a .< s < a Q t 1 i f a Q _< s _< b 47. j = l , 2 , . . . , n . Then i t i s o b v i o u s t h a t the p r o o f s of ( 3 . 1 l ) and (3 .12) ( w i t h b Q r e p l a c e d by b) h o l d f o r the p r e s e n t case . To p r o v e (3 .12) and (3.1*0 f o r the p r e s e n t case we note t h a t s i n c e ^ b ^ "^ 1J ^'n-m^ b i s f i x e d , •• •  • .< • i s constant , hence f r o m ( 6 . 6 ) i n e q u a l i -n b t i e s (3-12) and (3.1*0 are o b v i o u s on a D _< s _< b . The p r o o f t h a t these i n e q u a l i t i e s h o l d on a _< s < a Q f o l l o w s e a s i l y f r o m ( 3 - 3 ) . These i n e q u a l i t i e s . , i n a d d i t i o n t o ( 3 . 5 ) i m p l y the bounded-ness of (3 .1) by lemma 4. Hence we have the f o l l o w i n g r e s u l t s : Theorem 8. Assume t h a t tu_ i s a s i n g u l a r i t y f o r L as d e s c r i b e d i n t h i s s e c t i o n and t h a t c o n d i t i o n s ( 2 . 7 ) , ( 2 . 9 ) , ( 2 . 1 l ) , ( 3 - 3 ) , (3-5) and ( 3 . 6 ) are s a t i s f i e d . I f , i n a d d i t i o n , the m o n o t o n i c i t y p r o p e r t y of t h i s s e c t i o n i s s a t i s f i e d t h e n the e i g e n f u n c t i o n s x J c o r r e s p o n d i n g t o \ and c o r r e s p o n d i n g t o u„ of theorem 7 c t c l have the u n i f o r m r e p r e s e n t a t i o n y a ( s ) = x J ' ( s ) - f J ( s ) + 0(-0 a) , j * l , 2 , . . . , k on a < s < b p r o v i d e d I D_ < a <^ a Q where f J ( s ) i s the unique s o l u t i o n of the boundary p r o b l e m ( 6 . 7 ) L c f = 0 , U^f = U a x j , i = 1 , 2 , . . . , m IJ^f = 0 , i = 1 , 2 , . . . , n - m . The f o l l o w i n g v a r i a t i o n a l f o r m u l a e f o r X - as a - ID ) / a «• are immediate consequences of theorem 5. Theorem: 9. Under the assumptions of theorem 8 the f o l l o w i n g v a r i a t i o n a l f o r m u l a e f o r the e i g e n v a l u e s of theorem 7 are v a l i d : 4 8 . ( 6 . 8 ) X - u a - [ f V ] ( b ) - [ f V ] ( a ) + U 0 - \ ) ( f J , f J ' ) b as a tu_ where f J i s the unique solution of ( 6 . 7 ) , j = 1 , 2 , ...,k. 7. Class 1 singular problems fo r which a l l basic solutions are  i n H. Perturbation problems w i l l be considered for the case that the d i f f e r e n t i a l operator L given by ( 0 . l ) has s i n g u l a r i t i e s u)_,a>+ both of class 1 v a r i e t y and that the basic solutions are a l l In H. More p r e c i s e l y , we s h a l l assume that the d i f f e r e n t i a l equation L Qx = 0 has basic solutions q>1, i » l , 2 , . . . , n i n H such that (»_,tt)+ are not accumulation points of the zeros of cpi for each i , and such that j . i (s) ( 7 . 1 ) l i m ^ }s) = °» i = 1 . 2 , . . . ,m - 1 ; s -* uu_ i ^ ' C P ± ( S ) l i m 7—r = 0 , i = m+l,...,n Tl. a - a)+ TOi+l^sj The treatment i n t h i s section i s designed to include cases for which some of the basic solutions may be unbounded at OJ_ or tu+. As i n the previous cases we s h a l l define a basic eigenvalue problem on (OD_,ID+) and obtain estimates for the eigenvalues and eigenfunctions of the perturbed problem ( 1 . 3 ) for a,b near u)_,t»+. We s h a l l make use of the following lemma, which Is a generali-zation of Weyl's f i r s t theorem ( [ 9 ] , p.3 1 ) . 4 9 -Lemma 5 . Assume f o r some complex number a l l s o l u t i o n s of Lx = l^x are i n H [ a , u ) + ) , tu_ < a < uo+. Then f o r any complex number l, a l l s o l u t i o n s of Lx = are i n H [ a , t u + ) . P r o o f ; The p r o o f depends on the use of a v a r i a t i o n - o f - c o n s t a n t s f o r m u l a w h i c h d i f f e r s s l i g h t l y f r o m t h a t used by C o d d i n g t o n i n ( [ 2 ] , p . 1 9 5 ) . From ( 0 . 4 ) i t i s c l e a r t h a t [ x y ] ( s ) may be w r i t t e n i n the f o r m n - 1 "R. .. •3 [ x y ] ( s ) = £ B ± j ( s ) x ^ ( s ) y ( J ) ( s ) i , J = 0 w i t h ( 7 . 2 ) B±.(s) = ( - l ) j p 0 ( s ) , i + j = n - 1 0 , i + j > n - 1 L e t B denote the n - b y - n m a t r i x w h i c h has the element B ^ j i n the i + l - t h row and j + l - t h column, 1 , 3 = 0 , 1 , . . . , n - i . Then ( 7 . 2 ) i m p l i e s t h a t B i s n o n - s i n g u l a r on [a ,uu + ) . L e t cp^cpg* • • • be n l i n e a r l y independent s o l u t i o n s of Lx = l-^x. U s i n g G r e e n ' s f o r m u l a ( 0 . 5 ) we see t h a t [cpacpp](s) i s a c o n s t a n t [cp^cpp] independent of s. L e t S denote the m a t r i x w i t h element [cp^pJ i n the a - t h row and p - t h column, a , P = 1 , 2 , . Then i t i s e a s i l y v e r i f i e d t h a t ( 7 . 3 ) S = Y t B Y where Y denotes the Wronskian m a t r i x ( c p ^ " " ^ ( s ) ) , i , j = 1 , 2 , . . . and Y^ the t r a n s p o s e o f the m a t r i x Y. S i n c e the m a t r i c e s B , Y (and hence Y^) are n o n - s i n g u l a r on [a ,uo + ) , i t f o l l o w s t h a t S i s a n o n - s i n g u l a r c o n s t a n t m a t r i x . 50. Let = (y O ) denote the matrix inverse to S, and consider the function K of (s,t) defined by n ( 7 . 4 ) K(s,t) = £ Y a p q > P ( 8 ) < p a ( t ) . a, 0=1 For any closed subinterval [a,bj of [a , t » , ) l e t v e H[a,bJ. ¥e " T s h a l l show that the function b (7.5) u(s) = f K ( s , t ) v ( t ) k ( t ) d t J a i s such that u^11""1^ i s absolutely continuous on [a,b], and u s a t i s f i e s the d i f f e r e n t i a l equation (7.6) Lu = ^ 1 u + v . From (7.3) we have ( Y t ) ~ 1 S Y"1 = B and hence (7.7) Y S ' V = B " 1 . Let B^j denote the element i n the i + l - t h row and j+l-th column of B " 1 , i , j = 0,1,...,n-1. Then (7.2) c l e a r l y implies that -i i 0, i + J < n - 1 ( 7 - 8 ) B « - \ t l J i . Consequently from (7.7) and (7.8) i t follows that n (7'9) I V a pq> A(8)4 J )(8) - 0 a, 0=1 f o r j = 0,1,...,n-2, and 51. n a , (3=1 I t i s now a s t r a i g h t f o r w a r d c a l c u l a t i o n t o show f r o m (7.4), (7.5) t h a t u, u"^,. .. , u ( n ~ " ^ e x i s t and t h a t n ,s (7.11) u ( i } ( s ) . Y Y a p ^ p 1 ^ 8 ) I < P a ( t ) v ( t ) k ( t ) d t , a , 0-1 a i = 1 , 2 , . . . ,n-l. A l s o f r o m (7.10) we have (7.12) «<»)(.) = | Y a p 4 « ) ( s ) J % a ( t w t ) K ( t ) « + i i i ^ a ) a , p = l a ° From (7.11) and (7.12) i t i s now c l e a r t h a t u s a t i s f i e s (7.6). L e t x he any s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n Lx = lx w h i c h may be w r i t t e n i n t h e f o r m Lx = l^x + (l - l^)x. Then t h e v a r i a t i o n - o f - c o n s t a n t s f o r m u l a g i v e n b y (7.4) and (7.5) Y I 6 L D S * B / N \ (7.13) x(s) = £ c.cp.(s) + (i-^ )/ f £ Y a p«Pp(s)(r r t(t ) jx(t)k(t)dt i = l C \ a , p = l / where c, c^, Y ap a r e c o n s t a n t s . L e t M be a c o n s t a n t s u c h t h a t Y„ < M* P = 1,2,...,n where y n = max |Y„R|. ° p l < a , p X n a p The Schwarz i n e q u a l i t y g i v e s g / n \ n \ ( I W p ( a ) v a ( t ) ) x ( t ) k ( t ) d t < M J |<pp(s)| llxllc • c \ a , p = l / P=l U s i n g t h i s i n (7.13), t h e M i n k o w s k i i n e q u a l i t y y i e l d s 11x11* i f fkilV + n U - t j M 2 llxll* ' i - l 52. If c i s chosen large enough so that nl^—t^lM < then llxll c < 2 £|c.| M. i = l Since the r i g h t side of t h i s inequality i s independent of s i t follows that x e H[a,uj +). This completes the proof of the lemma. From lemma 5 i t i s clear that n independent end conditions are required to obtain a reasonable eigenvalue problem on (a> ,ti) +). To obtain suitable end conditions we s h a l l b a s i c a l l y follow the method suggested by Kodaira i n [6]. Our method w i l l also resemble those used by Coddington [2] and Weyl i n [9]. „ -1 Let D be the set of a l l x e H such that x e C (i» *u>+) (n—1) and x^ " ' i s absolutely continuous on every closed bounded sub-i n t e r v a l of (uo By an end condition at oo_ we s h a l l mean a condition for x € D of the form [x*](u)J = 0 where ijt i s a fi x e d function i n D such that L§ e H(iu_,a] for any a s a t i s f y i n g UJ_ < a < uu+. A f i n i t e number of end conditions at [ X , V 1 ] ( O J _ ) = 0, [ x t 2 ] ( o j _ ) = 0,..., [ x t k ] ( i » _ ) = 0 w i l l be c a l l e d l i n e a r l y independent at tw_ whenever -I a j[x,y . ] ( u ) J - 0 j - l i d e n t i c a l l y i n x implies a. =0, j = l,2,...,k. 5 3 . End c o n d i t i o n s at uu+ and t h e i r l i n e a r independence are t o be d e f i n e d s i m i l a r l y . L e t { 1 ^ , . . . , y\nY be any set of r e a l f u n c t i o n s i n D such t h a t L* . e H , j = 1,2, . . . , n and such t h a t the se t {[x*j](a)J = 0- J - 1,2,. . . ,£} i s l i n e a r l y independent at uo_, and the se t {[xtj]<a) + ) = 0, j = -TJ + 1,. . . ,n} i s l i n e a r l y i n d e p e n d e n t at uo . Then K o d a i r a ' s method a p p l i e d T t o the p r e s e n t case g i v e s r i s e t o the e i g e n v a l u e p r o b l e m ; Lx = \ x f o r a l l x e D such t h a t Lx e H and [ x f j = 0, j = 1,2,. . . ,^ , [x^fj](u) +) = 0, j = ^ + 1,.. . , n . A c t u a l l y K o d a i r a p u t s a d d i t i o n a l r e s t r i c t i o n s on the \\ . t o o b t a i n a s e l f - a d j o i n t p r o b l e m but we do not need the s e l f - a d j o i n t n e s s f o r our p u r p o s e s . F o r our purpose we s h a l l choose a se t of end c o n d i t i o n s w h i c h i s more g e n e r a l t h a n those demonstrated by K o d a i r a or Weyl . We s h a l l assume t h a t t h e r e e x i s t s n f u n c t i o n s X - ^ , . . . , " ) ^ i n D such t h a t L\ • e H , j = l , 2 , . . . , n and such t h a t the se t J { [ x X j ] ( t « _ ) = 0, j = l , 2 , . . . , m } i s l i n e a r l y independent at t»_ and the s e t {[XY.](U> +) = 0, j = m + l , . . . , n } 54. i s l i n e a r l y independent at w+, 1 _< m < n. Then the eigenvalue problem (7.14) Lx = xx, x e D, [xXj ]((»_) = 0 , j - l,...,m, [xxj](u> +) - 0, j = m+l,...,n w i l l be referred to as the basic problem and (uu_jU) +) as the basic  i n t e r v a l . Again we stress the point that (7.14) i s to be a reasonable problem,i.e. eigenvalues are supposed to exist. F i n a l l y we require that a l l eigenvalues of (7.14) are r e a l . This impliescthat l i s not an eigenvalue of ( 7.14). For the perturbed problem on [a,b], uu_< a < b < uu+ we choose the regular s e l f - a d j o i n t problem (1.3). The f i r s t theorem w i l l provide conditions under which the eigenvalues of (1.3) converge to those of (7.14) as a,b - t » _ , t t ) + . The remaining theorems are refinements which lead to asymptotic estimates of the eigen-values and eigenf unctions. Let X be an eigenvalue and {x., j = l,2,...,k] a cor-J responding orthonormal set of eigenfunctions for ( 7*14). Let A^ denote the space spanned by {x^, j « l,2,...,k}. To obtain convergence of the eigenvalues of (1.3) to those of (7.14) we assume that conditions ( 2 . 7 ) - (2.10) hold and that for any x e A x: (7.15) a ft m 1 = U o ( l ) a N a ™ i ) / 6-K( m +2,n) N\ ( 7 . 1 6 ) b 1 ' W l ' , o ( J ^ i j . o ( l ) as a,b - ® „ > w + f o r a H sequences ( i p i j n . i ) J'n-m-l^' 5 5 . Let (7.17) = sup X€A llxll-1 = sup x € Ax 1 'x lUl 6 a ( l , m - l ) fi. (m+2,n) Then ( 7.15) - ( 7 . 1 7 ) c l e a r l y imply that ( 7 . 1 8 ) p(a) = o ( l ) , p(b) = o ( l ) as a,b -* oo_,a)+. Assumptions (2.7) - (2.10) imply for the present case that ( 7 . 1 9 ) h ( a ) o a ( i , J ) - o ( l ) , h ( b ) a b ( i , j ) = o ( l ) for i = l,2,...,m, j = m+l,...,n and (7.20) (7.21) TT o ( l ) for (±lf±m) t (l,m), V Jl' Jn-m.) o( l ) for (J-L^n.m) ^ (m+l,n) as a,b -• . The weaker conditions ( 7 . 1 5 ) , ( 7 . l 6 ) , ( 7 . 2 0 ) , ( 7.21) are ac t u a l l y s u f f i c i e n t to obtain the convergence of the eigenvalues of ( 1 . 3 ) to those of ( 7.14)" while the stronger assumptions ( 2 . 7 ) - ( 2 . 1 0 ) , ( 7 . 1 5 ) , ( 7 . 1 6 ) . w i l l be required to obtain the uniform estimates i n theorem 1 2 . Theorem 10. Let U J_ and UB be s i n g u l a r i t i e s for L as described i n t h i s section. Let X be an eigenvalue of ( 7.14) possessing k orthonormal eigenfunctions. Then under assumptions ( 2 . 7 ) - ( 2 . 1 0 ) , 56. • ( 7 . 1 5 ) , ( 7 . 1 6 ) (or the weaker c o n d i t i o n s ( 7 . 1 5 ) , ( 7 . 1 6 ) , (7 . 20 ) , (7 .21) ) t h e r e e x i s t s a r e c t a n g l e R , and a c o n s t a n t C on R Q , such t h a t at l e a s t k p e r t u r b e d e i g e n v a l u e s of ( 1 . 3 ) s a t i s f y (7 .22) | u a b - x| < C(p ( a ) + p ( b ) ) whenever [a,b'] e R o P r o o f . L e t x be any n o r m a l i z e d f u n c t i o n i n A . Then , p r o c e e d i n g ~-" A. as i n theorem 1,- we d e f i n e a f u n c t i o n f on [ a , b ] by ( 7 . 2 3 ) f = x - Y G a b x where y = X - lQ. Then f i s the unique s o l u t i o n of the boundary p r o b l e m (7 .24) L f = 0, U*f = i f x , i = l , 2 , . . . , m , o a a TJ^f = U^x, i = 1 , 2 , . . . , n - m . I f the h ( ), Ti. ( ), 6 ( ), 6. ( ) n o t a t i o n i s used one may f i n d a r e p r e s e n t a t i o n of f i n terms o f the b a s i c s o l u t i o n s as f o l l o w s . . ti Y ^ 1 V ft /., N 6 a( V^ n - l ^ ^ b ^ ' l ^ ' n - m ^ (7 .25) f ( s ) - K j w r B ) l \ ) ( + ) i - -k = l N l a D p a D / where ^ a a n b where S I n d i c a t e s summation over a l l p o s s i b l e sequences ( i ^ , i m ) and ( J ^ J ' n ^ ) such t h a t ( i - ^ ^ ) £ (l*m) and such t h a t i r £ o'g, 5 7 . ( i i ) I i n d i c a t e s summation over a l l p o s s i b l e sequences ( i 1 , i m ^ 1 ) and ( J ^ J n ^ ) such th a t i r £ j g ^  k, r = 1 , 2 , . . . , m - 1 , s = 1 , 2 , . . .,n-m; ( i i i ) Z i n d i c a t e s summation over a l l p o s s i b l e sequences ( 2 ) ( i ^ ^ ) and ( J ^ J ^ n ^ ) such th a t i r ^  j g ^  k, r = l , 2,...,m, s - 1 , 2 , . . . , n - m - l . That t h i s r e p r e s e n t a t i o n ( 7 . 2 5 ) of f i s v a l i d f o l l o w s i n the same way as ( 2 . 2 1 ) . Prom ( 7 . 2 0 ) and ( 7 . 2 1 ) we can f i n d numbers a Q,b o, o> < a Q < b Q < O J + such that K(a,b) i s bounded away from 0 whenever <D_ < a _< a Q , b Q j< b < ID . Also by ( 7 . 1 5 ) - ( 7 . 1 7 ) , ( 7 . 2 0 ) , ( 7 . 2 1 ) there e x i s t numbers a Q , b Q (which may be pre-supposed to be the previous choices) such that , 0 ( p ) a b f o r a l l sequences ( i ] _ ^ i m ^ T _ ) a n d ( ^ 1 * J n , m _ 1 ) , a n d such that ~ T l % " ° < > b > f o r a l l sequences ( i ^ i ^ and ( J n _ m _ ] _ ) whenever ID < a < a ^ , b „ < b < t D , . These c o n s i d e r a t i o n s i n a d d i t i o n to — — o o —• + the f a c t t h a t cp . e H, j = l , 2 , . . . , n permit us to deduce that there J e x i s t s a re c t a n g l e R Q, and a constant C on R Q, such that 'a + pb whenever [a,b] e R . \ < < C ( p a + P h ) 58. By ( 7 . l 8 ) we have t h a t l l f l l a = o ( l ) as a,b -* uo_,uo+. Hence an a p p l i c a t i o n of lemma 1 (as i n theorem l ) shows t h a t t h e r e e x i s t s a c o n s t a n t C and a r e c t a n g l e R Q , such t h a t at l e a s t k e i g e n v a l u e s ( c o u n t i n g m u l t i p l i c i t i e s ) of (1 .3 ) s a t i s f y Kb - X l < C ^ a + Pb^ whenever [a,b] e R Q . T h i s completes the p r o o f of the theorem. Theorem 10 and (7.18) show i n p a r t i c u l a r t h a t f o r each b a s i c e i g e n v a l u e X of m u l t i p l i c i t y k t h e r e e x i s t at l e a s t k p e r t u r b e d e i g e n v a l u e s | j a b ( c o u n t i n g m u l t i p l i c i t i e s ) such t h a t l i a b -* X when a,b uo_,uo+. To o b t a i n the s t r o n g e r r e s u l t t h a t e x a c t l y k p e r t u r b e d e i g e n v a l u e s s a t i s f y (7.22) i n theorem 10 we r e q u i r e t h a t (7.1*+) s a t i s f y the f o l l o w i n g m o n o t o n i c i t y  p r o p e r t y : The a b s o l u t e v a l u e of t h e i - t h e i g e n v a l u e o f p r o b l e m (7 .14) i s not l a r g e r t h a n the a b s o l u t e v a l u e of the I - t h e i g e n v a l u e of (1 .3). The f o l l o w i n g theorem i s t h e n o b t a i n e d a n a l o g o u s l y t o theorems 2 and 3. Theorem 11. I f t h e m o n o t o n i c i t y p r o p e r t y h o l d s i n a d d i t i o n to the h y p o t h e s e s of theorem 10, t h e n f o r e v e r y e i g e n v a l u e X o f ( 7 . 1 4 ) , of m u l t i p l i c i t y k , t h e r e e x i s t s a r e c t a n g l e R Q and a c o n s t a n t C on R Q such t h a t e x a c t l y k p e r t u r b e d e i g e n v a l u e s | i a b o f (1.3) s a t i s f y tu£b - X| < C ( p a + p b ) , j = 1 , 2 , . . . , k whenever [a,b] e R Q . There e x i s t o r t h o n o r m a l e i g e n f u n c t i o n s x J a s s o c i a t e d w i t h X and y & b a s s o c i a t e d w i t h n a b such t h a t "^b - XK i C ( P a + pb> ' J - 1,2,...,k whenever [a,bj e R . o To obtain uniform estimates of y ^ b ( s ) - x J ( s ) i n theorem 11, stronger assumptions are needed on the behaviour of the basic solutions at uo_ and oo . In addition to the hypotheses of theorem 11, we s h a l l require that conditions (3.3) - (3.7) are s a t i s f i e d . Then the hypotheses of lemma 4 are c l e a r l y s a t i s f i e d and one can obtain the following theorems: Theorem 12. If i n addition to the hypotheses of theorem 11, conditions (3-3) - (3.7) are s a t i s f i e d , then the eigenfunctions x J corresponding to \ and y a b corresponding to u a b of theorem 11 have the uniform representation: y a b ( s ) = x J ( s ) - f J ( s ) + 0(pa + p b ) , j = l,2,...,k, for a s _< b, OJ_ < a _< a Q , b Q _< b < uu+ where f J ( s ) i s the unique solution of the boundary problem L f = 0 , ujf = uV" , I = l,2,...,m, u j f = U^x^, i = l,2,...,n-m. Theorem 13. Under the assumptions of theorem 12, the following v a r i a t i o n a l formulae hold for the eigenvalues X and u a b of theorem 10: X - n f b = i fV'Kb) - [fVj(a) +(i0 - x ) ( f J , f J ) ^ + K + pb)(fJ,D>(D as a,b uo , i u , for j = l,2,...,k. 60. 8. A one end p e r t u r b a t i o n problem; a l l b a s i c s o l u t i o n s i n H(uu , b] The m a t e r i a l i n the preceding s e c t i o n a p p l i e s e a s i l y to the case f o r which the b a s i c problem i s defined on (oj_,b], b f i x e d , b^ < b < in.. Hence we s h a l l consider the case that uo i s a o — •+• \ — c l a s s 1 s i n g u l a r i t y f o r L and a l l the b a s i c s o l u t i o n s are i n H(iu_,b]. Let D(u>_,b] be the set of a l l x e H(uo ,b] such that ( i ) x € C11""1" (uj_,b ] and x^11-"*") i s a b s o l u t e l y continuous on every closed bounded s u b i n t e r v a l of (w_,b]; ( i i ) x s a t i s f i e s the re g u l a r boundary c o n d i t i o n s Uj^x = 0 , i = 1 , 2 , . . . ,n-m where i s given by (0.7), and the end co n d i t i o n s [x%J.](uo_) = 0, j = 1 ,2 , . . . ,m of ( 7.14). Then the b a s i c problem to be considered i s the eigenvalue problem (8.1) Lx = Xx , x e D(«)_,b]. Again we r e q u i r e that at l e a s t one r e a l eigenvalue e x i s t s and that I is not an eigenvalue. We s h a l l compare problems ( 1 .3) and (8 .1 ) w i t h (1 .3) regarded as a p e r t u r b a t i o n of ( 8 . 1 ) . The re g u l a r endpoint b Ls to remain f i x e d f o r t h i s case. To o b t a i n the r e s u l t s corresponding t o theorems 10 and 11, we r e q u i r e that c o n d i t i o n s ( 2 . 7 ) , ( 2 . 9 ) , (7.15) (or even the weaker c o n d i t i o n s (7.15) and (7.20)) hold and that (8 .1) s a t i s f i e s the monotonicity property described i n s e c t i o n 6. We then have the f o l l o w i n g r e s u l t : 61. Theorem 14. Let UJ_ be a s i n g u l a r i t y for L as described In thi s section and l e t X be an eigenvalue of (8.1) possessing k ortho-normal eigenfunctions. Then, i f conditions ( 2 . 7 ) , ( 2 . 9 ) and (7.15) (or even the weaker conditions (7.15) and ( 7 . 2 0 ) ) are s a t i s f i e d and the monotonicity property holds, there exists an Interval (w„>&Ql and a constant C such that exactly k perturbed eigenvalues of (1 . 3 ) s a t i s f y c l - x i 1 °Pa > J = I* 2,...,k whenever u) < a < a . There exist orthonormal eigenf unctions — o ° x J corresponding to X and y£ corresponding to p.^  such that c l cL whenever u)_ < a _< a Q. To obtain uniform estimates on [a,b] and the v a r i a t i o n a l formulae for eigenvalues we assume i n addition to ( 2 . 7 ) , ( 2 . 9 ) , (7 .15) and the monotonicity property that conditions ( 3 . 3 ) , ( 3 . 5 ) and ( 3 . 6 ) hold. Then the boundedness of (3.1) on a a j£ s j< b, a _< a Q follows i n exactly the same way as In section 6. Theorem 15. I f , In addition to the assumptions of theorem 14, we require conditions ( 3 - 3 ) , ( 3 . 5 ) and ( 3 . 6 ) to be v a l i d then the eigenfunctions x^ corresponding to X and yt[ corresponding to St l i ^ of theorem 14 have the uniform representation y a ( s ) = x J ( s ) - f J ( s ) + 0 ( p a ) , j = 1,2,...,k for a _£ s _< b, UD_ < a _< a Q , where f J ( s ) i s the unique solution of the boundary problem b2. L Q f = 0 , U a f = u V , 1 = l , 2 , . . . , m , II* f = 0 , 1 = l , 2 , . . . , n - m . The f o l l o w i n g v a r i a t i o n a l f o r m u l a e are a l s o v a l i d : ( 8 . 2 ) X - n a - t f V ] ( b ) - [ f V ] ( a ) + ( l 0 . l ) ( f ; i , f J ) l a . + p a ( f J , l ) * 0 ( l ) as a -* O J _ f o r j = l , 2 , . . . , k . 9 . The second o r d e r c a s e ; U J _ , U J + c l a s s 1 l i m i t c i r c l e s i n g u l a r i t i e s . The assumptions and r e s u l t s of s e c t i o n 7 w i l l he s p e c i a l i z e d t o the second o r d e r case , i . e . t o the o p e r a t o r L d e f i n e d at the b e g i n n i n g of s e c t i o n 5- The case t o be c o n s i d e r e d i s t h a t f o r w h i c h "both o> and uu, are c l a s s 1 s i n g u l a r i t i e s and of the l i m i t c i r c l e v a r i e t y ( [ 3 J , p . 225). A theorem of Weyl ( [ 9 J * P- 3 9 ) s t a t e s t h a t t h e r e e x i s t l i n e a r l y independent s o l u t i o n s cp-^ , qp,^  e H of L Q x = 0 such t h a t [cp 1cp 1J(uj +) = [cp2cp2J(w_) = 0 and [cp 1 Cp 2 J(s) = 1 , (J)_ < s < t » + . I t w i l l be assumed t h a t are not a c c u m u l a t i o n p o i n t s f o r the z e r o s of cp-^  and cp2 . L e t D 2 denote the se t of a l l x e H such t h a t ( i ) x i s d i f f e r e n t i a b l e on ( U J _ , U D + ) and x ' i s a b s o l u t e l y c o n t i n u o u s on e v e r y c l o s e d bounded s u b i n t e r v a l of (uo ,i» ); 63. ( l i ) x s a t i s f i e s the end c o n d i t i o n s [ x c P 1 H t t ) + ) = [xcp2]|0 = 0. Then the e i g e n v a l u e p r o b l e m ( to be r e f e r r e d t o as the b a s i c problem) (9.1) Lx = Xx , x e D 2 i s known t o have a denumerable se t of r e a l e i g e n v a l u e s X^ and a c o r r e s p o n d i n g o r t h o r n o r m a l set of e i g e n f u n c t i o n s complete i n H. I t i s a l s o known t h a t each e i g e n v a l u e X^ of (9*1) has m u l t i p l i c i t y 1. The p e r t u r b e d p r o b l e m on [ a , b j , < a < b < uu+ t o be c o n s i d e r e d i s the r e g u l a r s e l f - a d j o i n t e i g e n v a l u e p r o b l e m g i v e n by (5-3). The s t r o n g e r assumptions of theorem 10 when a p p l i e d to the p r e s e n t case a r e : rpp(a) <P-,(b) U x [Ix (9-3) = o ( l ) ; uV = o ( l ) u acp 1 u f ecp2 as a ,b -» and t h a t (Q 41 ^ 2 r f l ( a ) V P j < P 2 ( P ) ^ ' j h ( a j rp 2 (a) ' h ( b j U b cp 2 V l ( b ; are bounded on neighbourhoods tu < a < a . b „ < b < m, of 0 — o o — + w^, uo+ r e s p e c t i v e l y . Assumptions (9-2) and (9-i0 i m p l y t h a t <p2(a) (b) 6 4 . ( 9 . 6 ) Ua?2 = o ( l ) V i = o ( l ) as a ,b - uo_,uj+ . The weaker c o n d i t i o n s of theorem 10 are p r e -c i s e l y ( 9 - 3 ) and ( 9 . 6 ) . A l s o a m o n o t o n i c i t y p r o p e r t y of ( 9 - l ) i s known t o h o l d ( f rom the maximum-minimum p r i n c i p l e f o r e i g e n -v a l u e s , [ 4 ] , [ 1 0 ] ) hence theorems 10 and 11 g i v e the f o l l o w i n g r e s u l t s w i t h a V Pb -V Theorem 16. L e t tu and no, he s i n g u l a r i t i e s f o r L as d e s c r i b e d — — — — — — — — "T I n t n i s s e c t i o n . L e t X and x be an e i g e n v a l u e and the c o r r e s -p o n d i n g n o r m a l i z e d e i g e n f u n c t i o n of ( 9 . 1 ) . I f c o n d i t i o n s ( 9 - 2 ) , ( 9 . 3 ) and ( 9 - 4 ) (or even the weaker c o n d i t i o n s ( 9 - 5 ) and ( 9 - 6 ) ) are s a t i s f i e d t h e n t h e r e e x i s t s a r e c t a n g l e R Q and a c o n s t a n t C on R Q such t h a t e x a c t l y one e i g e n v a l u e a. ^ of ( 5 - 3 ) s a t i s f i e s lab - X | < C U x a U x + V V 2 whenever [ a , b ] e R_. There e x i s t s a n o r m a l i z e d e i g e n f u n c t i o n y a b of ( 5 - 3 ) c o r r e s p o n d i n g t o u a > i such t h a t ab l y a b " x " a < C ' u a * l + ^2 whenever [ a , b ] e R Q . The assumptions i n theorem 12 when a p p l i e d t o the p r e s e n t case y i e l d , i n a d d i t i o n t o ( 9 . 2)- (9 .4 . ) , t h a t ( 9 . 7 ) h ( a ) cp 2(a) cp 1 (aj _< h ( s ) c p 2 ( s ) « P l ( s ) ( 9 . 8 ) h ( b ) cp 1(b) < h ( s ) rp]_ (S) qp 2 (bj cp 2(s j , oj_ < a _< s _< a Q , b < s < b < UJ o — — + 65. and t h a t (9.9) h ( s ) cp 1(s)||h(t) cp 2 (t)ir h ( s ) c p 2 ( s ) i l h ( t ) <p1(t)il are "bounded on ai < s < tu . We -then have the f o l l o w i n g r e s u l t s : "T" Theorem 17. Under assumptions (9.2)-(9.4), (9.7)-(9.9) the e i g e n f u n c t i o n s x and y ^ of theorem 1 6 have the u n i f o r m r e p r e s e n t a -t i o n : y a b ( s ) = x ( s ) - f ( s ) + 0( V V i + U b x V 2 on a _< s < b , D J _ < a _< a Q , b Q _< b < U J + where f i s the unique s o l u t i o n of the boundary p r o b l e m L f = 0 , U f = U x , U J = U. x . o 9 a a b b The f o l l o w i n g v a r i a t i o n a l f o r m u l a f o r the e i g e n v a l u e s X, p i s v a l i d as a , b -* U J _ , « J + : ab X - H a b = [ f x ] ( b ) - [ f x ] ( a ) + (lQ - X ) ( f , f ) a 66. CHAPTER I I CLASS 2 SINGULAR PROBLEMS P r e l i m i n a r i e s The v a r i a t i o n of the e i g e n v a l u e s and e i g e n f u n c t i o n s of the r e g u l a r s e l f - a d j o i n t e i g e n v a l u e p r o b l e m ( 1 . 3 ) w i l l be c o n s i d e r e d f o r a /b near ou_,uu i n the case t h a t a l l the b a s i c s o l u t i o n s cp^  are i n H and are s i n g u l a r i t i e s f o r L b o t h of c l a s s 2 v a r i e t y . A g a i n s u i t a b l e s i n g u l a r e i g e n v a l u e problems w i l l be d e f i n e d on u)_ < s < uu+ and c o n d i t i o n s w i l l be o b t a i n e d such t h a t the e i g e n -v a l u e s of ( 1 . 3 ) converge t o t h o s e of the s i n g u l a r e i g e n v a l u e p r o b l e m as a I D _ , b -* I D + . S i n c e a l l the b a s i c s o l u t i o n s are assumed t o be i n H , i t f o l l o w s f r o m lemma 5 t h a t n l i n e a r l y independent end c o n d i t i o n s w i l l be r e q u i r e d t o o b t a i n a s i n g u l a r e i g e n v a l u e p r o b l e m on (uo_,uj +). The c o r r e s p o n d i n g s i t u a t i o n f o r the second o r d e r o p e r a t o r i s t h a t f o r w h i c h b o t h U J _ and « j are l i m i t c i r c l e s i n g u l a r i t i e s a c c o r d i n g t o W e y l ' s c l a s s i f i c a t i o n ([3]j> p . 225). The t h e o r y of c l a s s 2 p e r t u r b a t i o n s w i l l be d e v e l o p e d i n s e c t i o n s 1 0 - 1 3 . The r e s u l t s w i l l be s p e c i a l i z e d i n s e c t i o n 14 t o second o r d e r o p e r a t o r s w i t h l i m i t c i r c l e s i n g u l a r i t i e s at uo . 67. 10. D e s c r i p t i o n o f the b a s i c and p e r t u r b e d problems. To e s t a b l i s h an e i g e n v a l u e p r o b l e m on (to__,uu+) f o r L as g i v e n by (0.1), we s h a l l b a s i c a l l y f o l l o w the method, suggested by K o d a i r a i n [6], (See a l s o [2] where mixed c o n d i t i o n s a re used..) L e t D be the s e t of a l l x € H such t h a t x s Cn~"'"((i) ) and x ( n ~ " ^ i s a b s o l u t e l y c o n t i n u o u s on e v e r y c l o s e d bounded sub-i n t e r v a l o f (cu_juu +). L e t X ^ i = l , 2 , . , . , n be a s e t of f u n c t i o n s ( t o r e m a i n f i x e d ) i n D such t h a t L X i e H, I = 1,2,.,.,n, and such t h a t the end c o n d i t i o n s [xXjJu>_) = 0, I = l,2,...,m are l i n e a r l y i n dependent a t uu_ and [xx 1](t« +) = 0, i = m+l,...,n are l i n e a r l y i ndependent a t U J + . Then t h e b a s i c p r o blem i s the s i n g u l a r e i g e n -v a l u e p r o b l e m (10.1) Lx = Xx , x e D Q 'where D I s t h e s e t of a l l x e D such t h a t o (10.2) f [ x X - J t i O = 0 , i = l , 2 , . . . , m I [ x X 1 ] ( u o + ) = 0 , i = m+l,...,n. A g a i n we s t r e s s t h a t (10.l) i s t o be a r e a s o n a b l e e i g e n v a l u e p r o b l e m , i . e . a t l e a s t one e i g e n v a l u e i s supposed t o e x i s t . A l s o we r e q u i r e t h a t a l l e i g e n v a l u e s o f (10.l) are r e a l . The p e r t u r b e d p r o blem i s the r e g u l a r s e l f - a d j o i n t e i g e n v a l u e p r o b l e m g i v e n by (1.3), and i s d e f i n e d f o r each [a,b] € R . F o r the c l a s s of p e r t u r b a t i o n problems t o be c o n s i d e r e d , the b a s i c s o l u t i o n s are not n e c e s s a r i l y o r d e r e d a c c o r d i n g t o t h e i r a s y m p t o t i c b e h a v i o u r at <H . C o n s e q u e n t l y s t r o n g e r c o n d i t i o n s have to be imposed on the l i m i t i n g b e h a v i o u r of the boundary o p e r a t o r s U, as a .b ~» uo ,uu,. In p a r t i c u l a r we s h a l l r e q u i r e t h a t f o r a o - + e v e r y n - l t i m e s d i f f e r e n t i a b l e f u n c t i o n y (10.3) f u a y = [ y x ± ] ( a ) [ l + o ( l ) ] as a - U J _ , i = 1,2,. . . ,m X y = [ y x m + 1 ] ( b ) [ l + o ( l ) ] as b - t » + , i = 1,2,...,n-m. L e t A denote the m a t r i x ( A . . ) where [cpj^Xj] (o>_ ), i = 1,2, . . . , n ; j = 1,2, . . . , m A. . = 1 J 1 [<p*X*] . ), 1 = 1,2-, . . . , n ; j = m + l , . . . , n and l e t fi = det A. Then s i n c e fi =,det A* where A* i s the t r a n s -pose of A and s i n c e I i s n o n - r e a l i t f o l l o w s i m m e d i a t e l y t h a t fi ^ 0 ( o t h e r w i s e I would be an e i g e n v a l u e of (10. l ) } . A l s o f o r each j , j = l , 2 , . . . , n , cp., Lcp . , L X . are i n H hence (0,5) i m p l i e s t h a t each l i m i t [ c p ^ X ^ ] ^ , ) e x i s t s (and i s f i n i t e ) f o r i , j = l , 2 , . . . , n . T h i s i m p l i e s t h a t fi is e q u a l t o some n o n -z e r o c o n s t a n t , L e t A ( a , b ) denote the m a t r i x (A. - ( a , b ) ) where A ± 1 ( a , b ) = U c^p i = 1,2,. . . , m; j = 1,2,. . . , n a j U y T ^ V - i ' 1 = m + l , . . . , n ; j = 1,2, . . . , n 69. and l e t Q ( a , b ) = det A ( a , b ) . S i n c e the l i m i t s of [cp.x.. 1 (a) and [cp .X . ] (b) are f i n i t e as a -* oo_ and b -• t» f o r i , j = l , 2 , . . . , n , i t f o l l o w s f r o m ( 1 0 . 3 ) t h a t we can s e l e c t numbers a 0 , b Q (which may be p r e - s u p p o s e d t o be the o r i g i n a l c h o i c e s ) and a c o n s t a n t C such t h a t (10 .4) f U^cp-j 1 < C , l u j c p j l < C f o r I , j = 1 , 2 , . . . , n whenever U J _ < a _< a Q , b Q _< b < uu . A l s o by ( 1 0 . 3 ) the element i n the i - t h row and j - t h column i n A ( a , b ) approaches the element i n the i - t h row and j - t h column i n A^ as a ,b -» uu ,uu, . T h i s i m p l i e s t h a t — + (10 .5) n(a,b) - 0 £ 0 as a ,b ~* and hence we can assume by (10.4) and (10.5) t h a t the numbers a 0 , b Q p r e v i o u s l y chosen are such t h a t Q ( a , b ) i s boxinded above and away f r o m z e r o whenever uu < a _< a Q , b Q _< b <. y j + . 11. ' Comparison of the b a s i c and p e r t u r b e d p r o b l e m s . The two problems ( 1 0 . l ) and ( 1 . 3 ) w i l l be compared, w i t h ( 1 . 3 ) r e g a r d e d as a p e r t u r b a t i o n of (10. l ) . We are g o i n g t o e s t i m a t e the v a r i a t i o n of the e i g e n v a l u e s and e i g e n f u n c t i o n s under the p e r t u r b a t i o n D Q -* D [ a , b ] and show t h a t t h i s v a r i a t i o n has the l i m i t 0 as a,'b -• u)_,uu+. L e t X be an e i g e n v a l u e of (10 .1) and l e t A denote the e l g e n s p a c e of d i m e n s i o n k c o r r e s p o n d i n g t o X. Let x , • i » x . , j = 1 ,2 , . . . , k be an o r t h o n o r m a l b a s i s f o r A, and l e t T (X) and T?"(X) be d e f i n e d by 70. k ( l l . D T a ( x ) = I | U a X j | , j = l k T J ( X ) - I l u j x j l j = l Then (10.2) and (10.3) c l e a r l y Imply t h a t (11.2) T a ^ x ) = ' 1 = 1 > 2 ^ - - > m ^ T J ( X ) = o ( l ) , I = 1 , 2 , . . . , n - m as a -* uu , h *-• uo . The f o l l o w i n g theorem p r o v e s the convergence of the e i g e n v a l u e s of (1.3) t o those of (10. l ) . Theorem 18. L e t uu_ and uu be s i n g u l a r i t i e s f o r L as d e s c r i b e d i n s e c t i o n 10. L e t X be an e i g e n v a l u e of (10.l) p o s s e s s i n g k o r t h o n o r m a l e i g e n f u n c t i o n s . Then under assumption (10.3) t h e r e e x i s t s a r e c t a n g l e R , and a c o n s t a n t C on R Q , such t h a t at l e a s t k p e r t u r b e d e i g e n v a l u e s u a b o f (1.3) s a t i s f y m n-m ( 1 1 . 3 ) - X| < C(l T * ( X ) i = l i = l whenever [a,"b] e R . P r o o f . P r o c e e d i n g as i n the p r o o f o f theorem 1 we d e f i n e a f u n c t i o n f on [ a , b J by (11 . 4 ) f = x - Y G a b x where y = \ - I and x e A ^ . Then f i s the unique s o l u t i o n o f the boundary p r o b l e m 71. L f = 0, uif = U*x, i = l,2,...',m 0 c t CX i * U^f = U^x, i = 1,2,...,n-m. Let K^(a,b) denote the determinant of the matrix obtained from A(a,b) by r e p l a c i n g the j - t h column by tJ-'-x , U 2x , . . . , U™x , ufx , . . . , U?" mx . a a a b J b Then f has the f o l l o w i n g r e p r e s e n t a t i o n i n terms of the b a s i c s o l u t i o n s : n (11.5) f(s) ~ -m±rs) I K J(a,b) ^ ( s ) . J=l Consider now the determinant K J(a,b) on the r e c t a n g l e , tu_ < a _< a Q , b Q _< b < Since each element of A(a,b) i s bounded on t h i s r e c t a n g l e "by (10.4), there e x i s t s a constant C such that m n-m • a • . - ^ X | -i — 1 i = l m n-m |K J(a,b)j < c ( £ |u|x| + 1 1 ^ f o r each j , j = l,2,...,n whenever co < a < a_, b < "b < uo . ™* — O O ™" "T" This a p p l i e d to (11.5) y i e l d s a constant C on R Q such that n / m n-m \ j = l 1=1 i = l But fi(a,b) i s bounded away from 0 on R Q by (10.5) and cp . e H, j = l , 2 , . . . , n j hence there e x i s t s a constant C such that m n-m ( 1 1 . 6 ) Hf||* < c(£|lftc| + I |ujx|) 1=1 i t a l ' holds uniformly on ou_ < a _< a Q , b Q < b < uu+. By ( 1 1 . l ) , (11.4) and (11.6) one may deduce that there e x i s t s a constant C such that 7 2 . for any normalized x e A. m n-m ( 1 1 . 7 ) l l x - Y G a b < < C ( l ^ ( x ) + Y T J(x)) l l < 1 = 1 1=1 whenever a)_ < a _< a Q, b Q _< h < I D + . It follows from ( 1 1 . 7 ) and lemma 1 that there exists a constant C on Rn such that o m n-m 1 = 1 i = l With the choice m n-m 6 i = l i = l we conclude that P(6) = 0 implies x = 0 on [a,b]. But dim A^ = k; hence there exist at least k perturbed eigenvalues u (counting m u l t i p l i c i t i e s ) of ( 1 . 3 ) such that i f [a,b] € RQ. This completes the proof of the theorem. Theorem l 8 and ( 1 1 . 2 ) show i n p a r t i c u l a r that i f X. i s a basic eigenvalue of m u l t i p l i c i t y k there exist at least k perturbed eigenvalues u a b (counting m u l t i p l i c i t i e s ) such that l i a b -» X when a,b ^ U J _ , O D + . To obtain the stronger r e s u l t that exactly k perturbed eigenvalues s a t i s f y ( 1 1 . 3 ) i n theorem l 8 , we require the monotonicity property that the absolute value of the i - t h eigenvalue of ( 1 0 . 1 ) i s not larger than the absolute value of the i - t h eigenvalue of ( 1 . 3 ) . We then have the following theorem: m n-m i= l i = l 7 3 . Theorem 19. If i n addition to the hypotheses of theorem l 8 the monotonicity property holds, then for every basic eigenvalue X. of ( 1 0 . l ) , of m u l t i p l i c i t y k, there exists a rectangle R Q and a constant C on RQ, such that exactly k perturbed eigen-values u a b (counting m u l t i p l i c i t i e s ) of ( 1 . 3 ) s a t i s f y m n-m i = l l = l whenever [a,b] € R . There ex i s t k orthonormal eigenfunctions x o associated with X and k orthonormal eigenfunctions y a b asso-ciated with |a a b such that m n-m i = l i = l j = 1 , 2 , . . . ,k whenever [a,b] € RQ. 1 2 . Uniform estimates and asymptotic v a r i a t i o n a l formulae. To obtain uniform estimates f o r y a b ( s ) " x ^ ( s ) i n theorem 1 9 , additional r e s t r i c t i o n s are needed on the basic solutions cpj, j = l , 2 , . . . , n . In p a r t i c u l a r we s h a l l require that a l l the basic solutions are bounded on (o>_,a>+). Lemma 6 . Let G a b ( s , t ) be the Green's function f o r k« L Q associ-ated with the boundary conditions of ( 0 . 6 ) . Then the p o s i t i v e function S g ^ s ) defined by ( 1 2 . 1 ) g a b ( s ) = j" |G b ( B , t ) - | 2 k ( t ) d t a 74. i s u n i f o r m l y bounded on a < s < b p r o v i d e d a _< a , b Q _< b . P r o o f . We f i r s t c o n s t r u c t the G r e e n ' s f u n c t i o n G a b ( s , t ) f o r k - L Q a s s o c i a t e d w i t h the boundary c o n d i t i o n s of ( 0 . 6 ) f o l l o w i n g the method used i n lemma 5 - D e f i n e a f u n c t i o n K a b ( s , t ) of ( s , t ) on [ a , b j by f K ( s , t ) , a < t < s < b (12.2) K a D ( s , t ) = 1 - - _ d , u L O > a _< s < t _< b where K ( s , t ) i s g i v e n by ( 7 . 4 ) . I t was shown i n lemma 5 t h a t f o r any v e H [ a , b ] the f u n c t i o n u g i v e n by r b r s u ( s ) = J K b ( s , t ) v ( t ) k ( t ) d t = J K ( s , t ) v ( t ) k ( t ) d t a a i s s u c h , t h a t u ( n _ 1 ^ ( s ) i s a b s o l u t e l y c o n t i n u o u s on [ a , b ] and u s a t i s f i e s the d i f f e r e n t i a l e q u a t i o n L Q u = v . L e t G b ( s , t ) be the f u n c t i o n of ( s^ t ) on [ a , b ] d e f i n e d by n (12.3) G a b ( s , t ) = K a b ( a , t ) +1 A k cp k (s ) k=l where the are chosen i n such a way t h a t G b ( s , t ) as a f u n c t i o n of s s a t i s f i e s the boundary c o n d i t i o n s of (0.6). Then c l e a r l y G a b ( s , t ) i s the G r e e n ' s f u n c t i o n f o r k - L Q a s s o c i a t e d w i t h the boundary c o n d i t i o n s of (0.6). A p p l y i n g these boundary c o n d i -t i o n s t o (12.3) and u s i n g C r a m e r ' s r u l e we o b t a i n t h a t ( 1 2 - * ) A k - A k ( t ) - - f c ) where fiab("t) denotes the d e t e r m i n a n t of the m a t r i x o b t a i n e d f r o m A ( a , b ) by r e p l a c i n g the k - t h column by m z e r o s n n o, o, o , I v±fl±{t)i%pj , , I y±f±(t)^y3 S i n c e cp^ . e H , k = l , 2 , . . . , n we o b t a i n i m m e d i a t e l y f r o m (10 . 6 ) and (10.7) t h a t t h e r e e x i s t s a c o n s t a n t C such t h a t (12.5) HAk(t)L < C > k - 1*2,... , n whenever a < a . b „ < b . — o ' o — I t f o l l o w s f r o m (12.1) t h a t s \ 1 b \ (12.6) g a b ( s ) < ^ | G a b ( s , t ) | 2 k ( t ) d t j ^ + ^ j G a b ( s , t ) | k ( t ) d t ) 1 "5 By (12.2), (12.3) and the t r i a n g l e i n e q u a l i t y we o b t a i n t h a t 1 n n s 1 a ' i , j = l j = l s \ n U l G a D ( s , t ) | 2 k ( t ) d t ) ^ < Y\y±j «PJ(B)| l l « P i ( t ) i i a + Il«pj(s)l l i A j C - t ) } ! But cp. i s bounded on (uo_,o) ) and rp. e H f o r each j , j = l , 2 , . . . , n ; hence by (12.4) ( and (12.5) the f i r s t q u a n t i t y on the r i g h t i n (12.6) i s u n i f o r m l y bounded on a _< s _< b p r o v i d e d a _£ a Q , — ^ s i m i l a r p r o o f shows t h a t the second i n t e g r a l on the r i g h t i n (12.6) i s a l s o u n i f o r m l y bounded on a _< s _< b p r o v i d e d a _< a Q , b Q _< b . T h i s g i v e s the d e s i r e d r e s u l t . We now s t a t e a theorem w h i c h g i v e s u n i f o r m e s t i m a t e s f o r the e i g e n -f u n c t i o n s of theorem 19. Theorem 20. I f i n a d d i t i o n t o the h y p o t h e s e s of theorem 19, rp. i s bounded on uu_ < s < oo , j = l , 2 , . . . , , n , t h e n the e i g e n -f u n c t i o n s x J c o r r e s p o n d i n g t o X and y a b c o r r e s p o n d i n g t o u a b of theorem 19 are such t h a t 76. m n-m (12.7) y a b ( s ) - x J ( s ) - H ( s ) + 0(1 T a ( x ) ) + o ( £ T J ( X ) ) , 1=1 1=1 j = l , 2 , . . . , k , where f J ( s ) i s the unique s o l u t i o n of the boundary p r o b l e m (12.8) L f = lQf , U a f = U a x J , i = l , 2 , . . . , m , ujf = U j x j ' , I = 1,2,. . . , n - m . Theorem 21. Under the hypotheses of theorem 20 the f o l l o w i n g v a r i a t i o n a l f o r m u l a e are v a l i d f o r the e i g e n v a l u e s u a b and \ of theorem 19: X - u a b = [ f V j ( b ) - [ f J x J ' ] ( a ) + (lQ - x ) ( f J ' , f J ) a m n-m ( i Ta<*> +l ^(«))(f J.DS 0(1) + i = l i = l as a ,b -• O J _ , ( D + f o r j = l , 2 , . . . , k where f J i s the unique s o l u t i o n of (12.8). 13. One end p e r t u r b a t i o n p r o b l e m s . As s p e c i a l cases of the p e r t u r b a t i o n p r o b l e m d i s c u s s e d i n s e c t i o n 10-12 we s h a l l c o n s i d e r problems where o n l y one end i s p e r t u r b e d w h i l e the o t h e r end remains f i x e d . " F o r t h i s purpose we s h a l l d e f i n e an e i g e n v a l u e p r o b l e m f o r L g i v e n by ( O . l ) on the i n t e r v a l [a,uo ), uo_ < a < uu+ . 77. L e t D[a,u) , ) denote the se t of a l l z e H[a,u},) h a v i n g the f o l l o w i n g p r o p e r t i e s : ( i ) z € Cn~^~[ a,uo +) and z ^ 1 1 - " ^ i s a b s o l u t e l y c o n t i n u o u s on e v e r y c l o s e d bounded s u b i n t e r v a l of [ a , u o + ) ; and ( i i ) z s a t i s f i e s the boundary c o n d i t i o n s of (0.6) at a and a l s o the end c o n d i t i o n s i n (10.2) at uo+. The e i g e n v a l u e p r o b l e m (13-1) Lz = vz , z e D[a,uo ) w i l l be r e f e r r e d t o as the s e m i - p e r t u r b e d problem' and may be r e g a r d e d as i n t e r m e d i a t e t o ( 1 0 . l ) and (1.3). The b a s i c assumptions r e g a r d i n g the s i n g u l a r i t i e s uo_ and ID, are as b e f o r e , i . e . we assume t h a t UD are c l a s s 2 s i n g u l a r i t i e s + + f o r L and t h a t a l l b a s i c s o l u t i o n s are i n H. A l s o we assume t h a t the complex number lQ, Im 1 ^ 0, i s not an e i g e n v a l u e of (13.1) f o r a i n some neighbourhood of uu_, say uu_ < a _< aNQ. Note t h a t an o r d e r i n g of the b a s i c s o l u t i o n s i s not assumed; hence we suppose t h a t tf^, have the l i m i t i n g b e h a v i o u r g i v e n by (10.3). (a) Comparison of problems (1.3) and (13. l ) . The e i g e n v a l u e problems (1.3) and (13. l ) w i l l be compared when uu_ < a _< a Q , a f i x e d , w i t h (13.1) r e g a r d e d as " b a s i c " and (1.3) r e g a r d e d as a p e r t u r b a t i o n of (13.1). We assume t h a t at l e a s t one r e a l e i g e n v a l u e v of ( 1 3 . l ) e x i s t s p o s s e s s i n g k o r t h o n o r m a l e i g e n f u n c t i o n s z L e t 78. j = i Then 'by (10 . 2 ) and (10.3) i t f o l l o w s t h a t (13 . 2 ) T J ( z ) = o ( l ) as b -* uu+ f o r i = 1,2, . . . , n - m . L e t A^ denote the space spanned by z . , j = l , 2 , . . . , k Then f o r any z e the f u n c t i o n f d e f i n e d on [ a , b ] by f = z - v G a b z , Y = v - lQ. s a t i s f i e s the boundary p r o b l e m L Q f = 0 , U a f = 0, I = 1,2,...,m, U^f = U ^ z , I = 1 , 2 , . . . , n - m . In terms of the b a s i c s o l u t i o n s , f has the r e p r e s e n t a t i o n Y K ^ ( a , b ) where K ^ ( a , b ) denotes the d e t e r m i n a n t of the m a t r i x o b t a i n e d f = L n ( a ,b) f r o m A ( a,b) Dy r e p l a c i n g the j - t h column b y 0, 0 , . . . , 0 , U ^ z , U 2 z , . . . , IJ^"mz . ^ l n ~ ^ e r o s Then an argument s i m i l a r t o t h a t used i n theorem l8 shows t h a t t h e r e e x i s t s a c o n s t a n t C and an i n t e r v a l b_ < b < uu. such t h a t o — + n-m 1 = 1 79. An a p p l i c a t i o n of lemma 1 now i m p l i e s t h a t at l e a s t k e i g e n v a l u e s l i ^ ( c o u n t i n g m u l t i p l i c i t i e s ) of (1.3) l i e on the i n t e r v a l n-m (13.3) |v - ^ | < c(l T £ ( Z ) ) i = l ' whenever b Q _< b < uo+. In p a r t i c u l a r (13.2) i m p l i e s t h a t t h e r e e x i s t at l e a s t k e i g e n v a l u e s (j^ -» v as b -» uo+. I f , i n a d d i t i o n , we r e q u i r e t h a t (13.1) s a t i s f i e s the m o n o t o n i c i t y p r o p e r t y , t h e n we o b t a i n e x a c t l y k e i g e n v a l u e s (j^ ( c o u n t i n g m u l t i p l i c i t i e s ) of (1.3) s a t i s f y i n g (13.3). In t h i s case one a l s o o b t a i n s o r t h o n o r m a l e i g e n f u n c t i o n s z° a s s o c i a t e d w i t h v and yjj a s s o c i a t e d w i t h the ^ such t h a t n-m i = l whenever b Q _< b < U J + , j = 1,2,. . . , k . To o b t a i n u n i f o r m e s t i m a t e s of y ^ ( s ) - z^'(s) on a _< s _< b and v a r i a t i o n a l f o r m u l a e f o r the e i g e n v a l u e s I J ^ as b - uu+ we need the a d d i t i o n a l assumption t h a t each ep. (s ) , j = l , 2 , . . . , n J i s bounded on some n e i g h b o u r h o o d of say b Q _< s < uu+. T h i s a s s u m p t i o n i m p l i e s t h a t each cp i s bounded on [a,tu ). Then I t i s e a s i l y shown t h a t the p o s i t i v e f u n c t i o n g a - b ( s ) d e f i n e d by (12. l ) i s u n i f o r m l y bounded on a _< s _< b , p r o v i d e d b _< b . The p r o o f of t h i s i s the same as t h a t of lemma 6 except f o r o b v i o u s s i m p l i f i c a t i o n s . The f o l l o w i n g u n i f o r m e s t i m a t e of y ^ ( s ) - z ^ ( s ) on a < s < b , b Q j< b , i s t h e n a d i r e c t consequence of theorem 20: 8o. n-m y ^ ( s ) = z J ' ( s ) - f J ( s ) + o(J T * ( Z ) ) , J = 1,2,...,* 1=1 where f J ( s ) i s the unique s o l u t i o n of the boundary p r o b l e m L o f = 0 > U a f = °> 1 = 1 > 2 > - ' * ' m U j f = U b z J " , i = 1,2,. . . , n - m . A l s o we o b t a i n as a consequence of theorem 21 the f o l l o w i n g v a r i a t i o n a l f o r m u l a e : v - ^ = [fV](l») - [ f J z J ] ( a ) + ( t D - x ) ( f J , f J ) J as b -» ID , j = 1,2, . . . , k . (b) Comparison of problems ( 1 0 . l ) and (13. l ) . We now compare problems (10. l ) and (13.l), w i t h (13.l) r e g a r d e d as a p e r t u r b a t i o n of (10. l ) . S i n c e the " p e r t u r b e d " p r o b l e m (13. l ) i s a s i n g u l a r p r o b l e m (on the h a l f open i n t e r v a l [ a , t D + ) ) s p e c i a l c o n d i t i o n s need t o be imposed t o o b t a i n the d e s i r e d e s t i m a t e s . L e t Ux denote the v e c t o r U a x , U^x, [ x x m + 1 j > + ) , [xx n](» +) We s h a l l assume t h a t t h e r e e x i s t s a number a Q , o> < a Q < ID , such t h a t the set of boundary c o n d i t i o n s Ux = 0 I s s e l f - a d j o i n t 8 i . whenever tu_ < a _< a Q , i . e . we r e q u i r e t h a t t h e r e e x i s t boundary forms U C J of r a n k n (See [ 3 L Chapter 11) such t h a t (13 .4) [uvJ(oo' ) - [ u v ] ( a ) = Uu.U^v + U c u . Uv h o l d s I d e n t i c a l l y i n u and v (here • r e p r e s e n t s the " d o t " p r o d u c t ) , Then ( 0 . 5 ) and ( 1 3 . 4 ) c l e a r l y i m p l y t h a t f o r any p a i r x , y e D [ a , u j + ) such t h a t L x , L y € H[a,uo ) ( L x , y ) a - ( x , L y ) a = 0. We s h a l l a l s o suppose t h a t (13.1) has r e a l e i g e n v a l u e s o n l y and a c o r r e s p o n d i n g set of e i g e n f u n c t i o n s complete i n H[a,uo ). T h i s w i l l be needed i n o r d e r t h a t lemma 1 w i l l a p p l y . S i n c e uo_ i s a c l a s s 2 s i n g u l a r i t y f o r L we r e q u i r e t h a t the boundary o p e r a t o r s U "^ have the l i m i t i n g b e h a v i o u r g i v e n by (10.3) as a -» iu . L e t G ( s , t ) denote the G r e e n ' s f u n c t i o n f o r k L 0 a s s o c i a t e d w i t h the boundary c o n d i t i o n s of (0.6) at a and the end c o n d i t i o n s (10.2) at MO and l e t G o be the l i n e a r t r a n s f o r m a t i o n on H[ a, IM, ) d e f i n e d by A G u ( s ) = G Q ( s , t ) u ( t ) k ( t ) d t , u e H [ a , ( « . ). a o „ a + a Let \ be an e i g e n v a l u e of ( 1 0 . l ) p o s s e s s i n g k o r t h o n o r m a l e i g e n f u n c t i o n s x j = l , 2 , . . . , k and l e t A denote the space J A . spanned by t h e s e e i g e n f u n c t i o n s x . . F o r any n o r m a l i z e d x e A.. J K we d e f i n e a f u n c t i o n f on [a,uo + ) by (13-5) f ( s ) = x ( s ) - yG x ( s ) where y = X - lf C L v 82. P r o c e e d i n g as i n s e c t i o n 11 we o b t a i n the f o l l o w i n g r e p r e s e n t a t i o n of (13 .5) i n terms of the b a s i c s o l u t i o n s : f(8) - I £ l S l ^ ( B ) k=l ° ( a ) where fi(a) i s the d e t e r m i n a n t of the m a t r i x A(a) d e f i n e d by A(a) = where hi - < r U^rpj i f i = l , 2 , . . . , m ; J = l , 2 , . . . , n C V l X . ] ( « ) . ) i f i = m + l , . . . , n ; j=l,2. j 1 1 ,n and where fi^(a) i s the d e t e r m i n a n t of the m a t r i x o b t a i n e d f r o m A(a) by r e p l a c i n g the k - t h column by U a x > ••••» ^ax> ^ x x m + l ^ U ) + ^ t x * n ] (<»+)• Prom (10 .3) and the f a c t t h a t Q ^  0 f o l l o w s t h a t t h e r e e x i s t s a p o i n t aQ, a)_ < a Q < uu+ (which may be p r e - s u p p o s e d t o be the o r i g i n a l c h o i c e ) such t h a t fi(a) i s bounded away f r o m 0 on oo < a-_< a . A l s o each element of A(a) i s bounded on <«_ < a <_ a c by (10.4) hence fik(a) = o ( £ T a ( x ) j i = l whenever uo_ < a _< a Q . S i n c e T k € H, k = l , 2 , 0 . . , n one deduces t h a t t h e r e e x i s t s an i n t e r v a l (uu_,a ] and a c o n s t a n t C such t h a t f o r any n o r m a l i z e d e i g e n f u n c t i o n x e A m A. f"a i C(ITa<*>) i = l whenever uo < a _< a Q 83. S i n c e (13-1) has a se t of e i g e n f u n c t i o n s complete i n m 1 H[a,uu + ) and s i n c e £ T a ( x ) "* 0 a s a ~* u u - b y ( 1 0 - 2 ) J ( 1 0 - 3 ) i = l and (11 . l ) we may a p p l y lemma 1 t o deduce the e x i s t e n c e of an i n t e r v a l («>_., a ] and a c o n s t a n t C on t h i s i n t e r v a l such t h a t at l e a s t k e i g e n v a l u e s v f ( c o u n t i n g m u l t i p l i c i t i e s ) of (13.1) s a t i s f y CL m (13.6) |x - v a | < -c (M;(x)) 1=1 whenever uo_ < a <_ a Q . I f i n a d d i t i o n the m o n o t o n i c i t y p r o p e r t y of ( 1 0 . l ) ( g i v e n i n s e c t i o n 11) i s assumed, t h e n one can show t h a t e x a c t l y k e i g e n v a l u e s ( c o u n t i n g m u l t i p l i c i t i e s ) of c l (13.1) s a t i s f y (13 .6). Under t h i s assumption one a l s o o b t a i n s n o r m a l i z e d e i g e n f u n c t i o n s x^ c o r r e s p o n d i n g t o X. a n d ' z ^ c o r r e s -61 p o n d i n g t o such t h a t c l m 11*J - z a l 1 a - C(I T a ^ x ) ) > J" = 1 > 2 > - - > * i = l wheneyer uo_ < a _< a Q . To o b t a i n u n i f o r m e s t i m a t e s f o r z^ - x J on [ a , b ] and c l v a r i a t i o n a l f o r m u l a e we s h a l l need the s t r o n g e r assumptions t h a t cp j ( s), j = l , 2 , . . . , n i s bounded on uo_ < s < c o + . The e x p l i c i t f o r m of G ( s , t ) can t h e n be o b t a i n e d by a p p l y i n g the boundary c l c o n d i t i o n s l ^ x = 0, i = 1 ,2 , . . . ,m and the end c o n d i t i o n s c l [ x X j ^ ] ( « > ' ) = 0, i = m + l , . . . , n t o n G a ( s , t ) = K a ( s , t ) + l A.cp. (s ) J=l as a f u n c t i o n of s where K ( s , t ) i s g i v e n by (12.2) w i t h b 84. r e p l a c e d by ID . S i n c e cp . € H and cp . i s bounded on (cu_,ou ), j = l , 2 , . . . , n , a p r o o f s i m i l a r t o t h a t of lemma 6 shows t h a t the p o s i t i v e f u n c t i o n g Q ( s ) d e f i n e d by g a ( s ) = J | G a ( s , t ) | 2 k ( t ) d t a 'I i s u n i f o r m l y bounded on a _< s < uo+, p r o v i d e d a _< a.Q. Then one may o b t a i n the f o l l o w i n g u n i f o r m e s t i m a t e s on a < s < o j , , a < a : . ~ ° M [(B ) = x d ( s ) - f J ( s ) + o ( J T a ( x j ) , j = 1,2,...,k i = l where f J ( s ) i s the unique s o l u t i o n of the boundary p r o b l e m V = 0 ' U a f = U a x J " ' 1 = I * 2 * . . - * ™ , [ f X i ] ( « ) + ) = 0 , i = m + l , . . . , n . The f o l l o w i n g v a r i a t i o n a l f o r m u l a e can a l s o be shown t o be v a l i d as a -> u ; \ - v a = [ f J x J ] ( a > + ) - [ f J x J ' ] ( a ) ; + ( t 0 - x ) ( f J , f J ' ) a m + (£^(x))(fJ,l) aO(l) , i = l j = 1 , 2 , . . . , k. 14. The second o r d e r c a s e ; uu_, t D + c l a s s 2 l i m i t c i r c l e s i n g u -l a r i t i e s . We c o n s i d e r as a p a r t i c u l a r case of ( 0 . l ) the o p e r a t o r 85. L = Lg d e f i n e d i n s e c t i o n 5. The p o i n t s a>_ and o>+ are i n g e n e r a l l i m i t c i r c l e s i n g u l a r i t i e s f o r L ; the p o s s i b i l i t y t h a t t h e y h e + 0 0 i s not e x c l u d e d . The n o t a t i o n s ( 0 . 3 ) and (0.4), w i l l be adhered t o ; i n p a r t i c u l a r (0 .4 ) t a k e s the f o r m (14 .1) [ x y ] ( s ) = p(s ) (x (s ) -y - rri7 - x ' ( s ) y(T7). The b a s i c p r o b l e m c o r r e s p o n d i n g t o ( 1 0 . l ) i s d e s c r i b e d as f o l l o w s : choose a complex number lQ, Im lQ ^ 0, and l e t L Q be the d i f f e r e n t i a l o p e r a t o r L - tQ. A theorem of Weyl ( [ 9 ] , PP. 35-44) s t a t e s t h a t t h e r e e x i s t l i n e a r l y independent s o l u t i o n s c p - j ^ c p 2 € H of L Q x = 0 such t h a t (14 .2) [ c p ^ ] ( a ) _ ) = [ c p 2 c p 2 ] ( u > + ) = 0 ' f c p ^ H s ) = 1 t U0_ < S < ( J U + . F o r our p u r p o s e s here we s h a l l assume o n l y the boundedness of cp and c p 2 o n ( O J _ , « J + ) . A c o n d i t i o n l i k e ( 5 . 4 ) r e g a r d i n g the " o r d e r i n g " of c p - ^ s ) and c p 2 ( s ) as s I D + i s not assumed. S i n c e ["cpgcPi] (o>_) ^ 0 and [ c p - ^ ] ( U J + ) ^ 0 by ( l 4 . l ) and (14 .2) we can choose X-^  and x^' ( d e s c r i b e d i n s e c t i o n 10) to be cp-^  and cp2 r e s p e c t i v e l y . L e t D 2 denote the set of a l l x e H w h i c h have the f o l l o w i n g p r o p e r t i e s : ( i ) x i s d i f f e r e n t i a b l e on (ai_,uo+) and x' i s a b s o l u t e l y c o n -t i n u o u s on e v e r y c l o s e d bounded s u b i n t e r v a l of (u)_,uo +); and ( i O [ x c p 1 ] (uu_) = [xcp 2 ](uj + ) = 0 . 86. The basi ' c e i g e n v a l u e p r o b l e m (14.3) Lx = Xx x e D 2 i s known t o have a denumerable se t of r e a l e i g e n v a l u e s {X*} and a c o r r e s p o n d i n g set of e i g e n f u n c t i o n s [ x i ] complete i n H ( i = 1 , 2 , . . . ). I t i s a l s o known t h a t each e i g e n v a l u e X has m u l t i p l i c i t y 1. The p e r t u r b e d p r o b l e m on [ a , b ] c o r r e s p o n d i n g t o (1.3) i s the r e g u l a r s e l f - a d j o i n t e i g e n v a l u e p r o b l e m g i v e n by ( 5 . 3 ) , A c c o r d i n g t o (10.3) we o b t a i n convergence of the e i g e n -v a l u e s and e i g e n f u n c t i o n s of (5.3) t o those of ( l 4 . 3 ) i f we r e q u i r e t h a t U , tl have the l i m i t i n g b e h a v i o u r cl D (14.4) f u a y = [ y c p 1 ] ( a ) [ l + o ( l ) ] as a - uo_ U b y = [ y c p 2 ] ( b ) [ l + o ( l ) ] as b - o>+ f o r e v e r y d i f f e r e n t i a b l e f u n c t i o n y : Let X be an e i g e n v a l u e f o r (14.3) and x the c o r r e s -p o n d i n g n o r m a l i z e d e i g e n f u n c t i o n . S i n c e e x a c t l y one boundary c o n d i t i o n i s used at each p o i n t a and b i n (5-3) we can r e p l a c e the q u a n t i t i e s m n-m i = l i = l i n (11. l ) by |U x| 3 \\J^x\ r e s p e c t i v e l y . F i n a l l y a m o n o t o n i c i t y p r o p e r t y f o r ( l 4 . 3 ) i s known t o h o l d [ 4 ] , [10], w h i c h l e a d s t o the f o l l o w i n g theorem: 87. Theorem 22. ( i ) Let uo__ and uu+ be l i m i t c i r c l e s i n g u l a r i t i e s for Lg. If the boundary operators U , U b s a t i s f y (14.4) then for each eigenvalue X of (14.3) there exists a rectangle R Q and a constant C on R c such that a unique eigenvalue u a b of (5-3) l i e s i n the i n t e r v a l l^ab - X ' < C d V l + I V ) whenever [a,b] e RQ'. There exist normalized eigenf unctions yab x associated with |a a b and X respectively such that the estimate i i y a b - < < c ( i V i + l V D i s v a l i d on R . ( i i ) I f , i n addition to the hypotheses of ( i ) , cp-^  and cp2 a r e bounded on (tw ,uo+) then the following uniform estimate i s v a l i d f o r a _< s _< b, [a,b] € RQ; y a b ( s ) = x(s) - f ( s ) + 0(|U ax| + |Ubx|) where f ( s ) i s the unique solution of the boundary problem L f = 0 , U f = U x , U, f = Tl x . o a a b b ( i i i ) Under the hypotheses of ( i i ) the following v a r i a t i o n a l formula i s v a l i d as a,b - uu_,u)+; X ' ^ ab = - [ f x ] ( a ) + (lQ - \)(t,t)\ + (|Uax| + |U bx|)(f,l)^0(l) . 88. CHAPTER I I I EXAMPLES 15. P r e l i m i n a r y remarks and lemmas. In t h i s c h a p t e r examples w i l l be g i v e n t o i l l u s t r a t e the the.ory of c h a p t e r I . The o p e r a t o r s t o be c o n s i d e r e d w i l l a l l be of the f o u r t h o r d e r w i t h c l a s s 1 s i n g u l a r i t i e s . Examples of problems i n v o l v i n g second o r d e r o p e r a t o r s are f o u n d i n [13]. We s h a l l need the f o l l o w i n g lemmas: Lemma 7 . L e t rp^> c p 2 , • . • 3 c p n be l i n e a r l y independent s o l u t i o n s 2n of c l a s s C of Lx = \ x where L i s the o p e r a t o r ( 0 . l ) and \ ^ 0, and l e t x.^ , X 2 , • • • 3 X n D e l i n e a r l y Independent s o l u t i o n s of 2n c l a s s C of Lx = - X x . Then t p 1 , r p 2 , . . . , c p n , X]_* X 2 * • — > X n 2 are l i n e a r l y independent s o l u t i o n s of LLx = X x. 2 P r o o f : F o r i = 1, 2, n, y^, Xj_ are s o l u t i o n s o f LLx = X x 2 2 s i n c e L L c p ± =L ( " X c p 1 ) = X cp± and L L X J _ = L ( - X X 1 ) = X \ ± . -n 1 = 1 ( ^ 1 + B i X ± ) = 0* t h e n n n ^ ( A 1 l 4 ) i + B ± L X 1 ) = ^ ( A ^ J L - B±x±) = 0. 1=1 1=1 Hence 2,1=1 A±V± ~ Xi=l B i * i = ° * w n i c n i m p l i e s A^ = = 0, i = l , 2 , . . . , n by h y p o t h e s i s . Thus cp-^ cp 2 , . . . , ^ X ^ X2> • — 3 X n are l i n e a r l y i n d e p e n d e n t . Lemma 8. I f c p ^ , c p 2 , . . . , c p n are l i n e a r l y independent s o l u t i o n s of Lx = 0 of c l a s s C and i f x ^ X 2 * X n are c o r r e s p o n d i n g s o l u t i o n s of c l a s s C of Lx = c p ^ , i = l , 2 , . . . , n , t h e n 89. ep]_* cpg, •. •, rp n , X-j_, X 2 , • X R are l i n e a r l y independent s o l u -t i o n s f o r LLx = 0. l 6 . The m o d i f i e d Hermite o p e r a t o r . As an example t o i l l u s t r a t e the t h e o r y of s e c t i o n s 1-4, c o n s i d e r the o p e r a t o r L = L Q L 0 on the I n t e r v a l ( - o o j 0 0 ) where L Q i s the m o d i f i e d Hermite o p e r a t o r g i v e n by L Q x = - x " + ( s 2 + 2)x. Then Lx = x ^ 4 ) - 2 ( s 2 + 2)x" - 4sx< + ( s 4 + 4 s 2 + 2)x o r , i n s e l f - a d j o i n t f o r m , Lx = ( x " ) n - 2 [ ( s 2 + 2 ) x ' ] ' + ( s 4 + 4 s 2 + 2)x . By (0.4) one o b t a i n s f o r L t h a t • (16..1) [ x y ] ( s ) = x ( 3 ) ( s ) y ( s ) - x " ( s ) y f ( s ) + x ' ( s ) y " ( s ) - x ( s ) y ( 3 ) ( s ) + 2 ( s 2 + 2)[x(si) y ' ( s ) - x ' ( s ) y ( s ) ] I t i s known ([15], p p . 347-348) t h a t t h e r e e x i s t l i n e a r l y independent s o l u t i o n s u and v of L n x = 0 such t h a t (16.2) u ( s ) as s -* - <=, and (16.3) u ( s ) ~ C e ^ ' s 1 / 2 , v ( s ) ~ C e ' ^ ' V 5 / 2 ~ C e ^ ' V ^ 2 , V (B ) ~ C e s 2 / 2 s 1 / 2 as s -• =0 90. Since the c o e f f i c i e n t of the f i r s t derivative term i s zero for L 0 i t follows that the Wronskian for L Q x = 0 i s constant and hence we can assume without loss of generality that (16.4) u(s)v'(s) - u'(s)v(s) = 1. Obviously u and v are l i n e a r l y independent solutions of Lx = 0. By lemma 8 two other l i n e a r l y independent solutions may be found by solving the d i f f e r e n t i a l equations Lx = u and Lx = v. The respective solutions are s s u( t ) v ( t ) d t - v( s ) f [ o J o fi S r S ^ (16.5) u(s) J u(t)v(t)dt - v(s)J [u(t)]2dt, and (16.6) -v(s)Pu(t)v(t)dt + u(s)f ""[v(.t)] 2dt . o o Let cp^  denote the function (16.5)» cp2 = u, cp-^  = v and l e t cp^  denote the function (16 .6). Then from ( 1 6 .2)-(l6. 6) one obtains that 2 2 (16.7) ~ Ce s / 2 | s | 1 / 2 l o g | s | , q>4 ~ Ce" s / 21 s | " 3 / 2 l o g | s | as s -» - o o , and (16.8) cp1 ~ Ce" s / / 2 s " 5 / 2 l o g s, cp4 ~ Ce s / / 2 s 1 / 2 l o g s as s - * . Clearly for any number c, -°° < c < 0 0 > ip 1, cp2 e H[c,») , Cp 1, cp2 4 H(-»,c], cp^, cp^  e H(-»,c], tpy cp^  4 H[c,«) and condition ( l . l ) i s s a t i s f i e d . By the asymptotic behaviour of tp^, i = 1,2,3,4 at + 0 0 one e a s i l y sees that the d i f f e r e n t i a l 91. e q u a t i o n Lx = 0 has no s o l u t i o n i n H = H ( - » , » ) . Hence l i s r e p l a c e d by 0 and the s o l u t i o n s cp^ i = 1,2,3,4 are r e g a r d e d as the b a s i c s o l u t i o n s . The b a s i c problem, on ( - » , » ) i s the . e i g e n v a l u e , p r o b l e m (16.9) Lx = Xx , x e D where D i s the set of a l l x e H such t h a t x e c P ( - » , » ) and x ^ i s a b s o l u t e l y c o n t i n u o u s on e v e r y c l o s e d bounded s u b i n t e r v a l of ( - 0 0 , 0 0 ) . By c o n s i d e r a t i o n of the o p e r a t o r L Q on (-00 , 0 0 ) one can deduce t h a t the e i g e n v a l u e p r o b l e m (16.9) has e i g e n v a l u e s 2 X n = (2n + 3) > n = 0,1,2, . . . f and c o r r e s p o n d i n g n o r m a l i z e d e i g e n f u n c t i o n s (16.10) x n ( s ) = 7 r - 1 / V ( n + 1 ) / 2 ( n j ) " 1 e x p ( - s 2 / 2 ) H n ( s ) , n = 0,1,..., where H n ( s ) denotes a Hermite p o l y n o m i a l . The w e l l - k n o w n a s y m p t o t i c b e h a v i o u r of x n ( s ) as s -» + 00 i s (16.11) x n ( s ) ~ i r - ^ 2 ( n + l ) / 2 ( n , r l / 2 8 n e 3 q p ( - 8 2 / 2 ) / 2 We now show t h a t X n = (2n+3) , n = 0,1,...j are the o n l y e i g e n v a l u e s of (16.9). S i n c e L i s f o r m a l l y s e l f - a d j o i n t , a l l e i g e n v a l u e s of (16.9) are n e c e s s a r i l y r e a l . Let n be any complex number such t h a t n ^ 0,1,2,.... Then i t i s known ([15] p p . 347-348) t h a t L Q x = (2n+3)x has l i n e a r l y independent s o l u t i o n s 'It-^ ijig such t h a t 2 2 / - I A , r<^s /2 - n - l , n ~ ~ s /2 n ( lb.12) ~ Ce ' s , i|tg ~ Ce ' s as s -• 0 0 , and 9 2 . ( 1 6 . 1 3 ) t l ~ C e - s 2 / 2 | s | n , t 2 ~ Ce^lsr*1-1 as s -* - 0 0 . S i m i l a r l y the e q u a t i o n L Q x = - ( 2 n + 3 ) x has l i n e a r l y independent s o l u t i o n s X J J X 2 such t h a t (16 .14) X l ~ C e - s 2 / 2 s - n - 5 , x 2 ~ C e s 2 / 2 s n + 2 as s -» « , and ( 1 6 . 1 5 ) X l - C e s 2 / 2 s n + 2 , x 2 ~ C e - s 2 / 2 s - n - 5 as s -* - » By lemma 7 , ty-^j ty2* X ] _ , X 2 a r e l i n e a r l y independent s o l u t i o n s of Lx = (2n+3) x and hence by ( 1 6 . 1 2 ) - ( 1 6 . 1 5 ) one can 2 deduce t h a t no s o l u t i o n of Lx = (2n+3) x i s i n H. T h i s i m p l i e s t h a t a number k i s an e i g e n v a l u e of ( 1 6 . 9 ) i f and o n l y i f 2 k = (2n+3) f o r some n o n - n e g a t i v e i n t e g e r n . A s i m i l a r p r o c e d u r e 2 shows t h a t t h e s e e i g e n v a l u e s X = (2n+3) > n = , 0 , 1 , . . . , a l l have m u l t i p l i c i t y 1. L e t D [ a , b ] be the set of a l l y e H [ a , b ] , - » < a < b < » such t h a t ( i ) y e C ^ [ a , b ] and y ^ ^ ( s ) i s a b s o l u t e l y c o n t i n u o u s on [ a , b ] ; ( i i ) Ly e H [ a , b ] ; and ( i i i ) y s a t i s f i e s the boundary c o n d i t i o n s U a y = U^y = 0 , I = 1 , 2 where ( 1 6 . 1 6 ) IJ^y = y ( 3 ) ( s ) - 2 ( s 2 + 2 ) y ' ( s ) U 2 y = y " ( s ) + s 5 y ' ( s ) . 93. Then the p e r t u r b e d p r o b l e m c o r r e s p o n d i n g t o (1.3) i s the s e l f -a d j o i n t e i g e n v a l u e p r o b l e m (16.17) Ly » (iy , y e D [ a , b ] , By a c t u a l c a l c u l a t i o n u s i n g (16.2) - (16.8) one can show t h a t 2 f- a (16.18) h ( 1 , 2 ) , ~ C e a a D ; h a ( 2 , 3 ) ~ Ca ; h a ( l , 3 ) ~ C a 4 l o g | a | ; n a ( 2 , 4 ) ~ C a 4 l o g | a l ; h a ( l , 4 ) ~ C a 4 ( l o g | a | ) 2 ; n a ( 3 , 4 ) ~ C e _ a a 4 as a -> - 0 0 , and 2 (16.19) h b ( l , 2 ) ~ C e " b b 4 ; h b ( 2 , 3 ) ~ C b 4 ; U b ( l , 3 ) ~ C b 4 l o g b j h b ( 2 , 4 ) ~ C b 4 l o g b ; 2 h b ( l , 4 ) ~ C b 4 ( l o g b ) 2 ; h b ( 3 , 4 ) ~ C e b b 6 as b -• «> . A l s o , i t i s e a s i l y v e r i f i e d t h a t (16.20) C 1 ( s ) = cp 3 (s) ; C 5 ( s ) = - c p 1 ( s ) ; C 2 ( s ) = cp 4 (s) ; C^(s) = - c p 2 ( s ) . By (16.10) and (16.16) one o b t a i n s f o r the e i g e n f u n c t i o n x n of (16.9) t h a t (16.21) I ^ x n ~ x ( s ) [ s 3 + (n+7)s] , U 2 x n ~ x ( s ) [ - s 4 + ( n + l ) s 2 ] as s -• + 00 and hence t h a t c o n d i t i o n s (2.11) and (2.12) are s a t i s f i e d w i t h 94 2/2 i Q i n - l / 2 (16.22) -©a = 0 [ e " a | a | l o g |a|] , a - — , 2/2 •9b = 0 [ e " b b n _ 1 / 2 l o g b] , b - co A l s o by (16.2) - ( l 6 . 8 ) , ( 1 6 . 1 8 ) , (16.19) and the a s y m p t o t i c b e h a v i o u r of the cp ,^, i = 1 , 2 , 3 , 4 i t i s e a s i l y seen t h a t c o n d i -t i o n s (2 .7) - ( 2 . 1 0 ) , (3 .3) - (3 .7) are s a t i s f i e d i f we choose h ( s ) = r 1 ' s l 1 ^ 2 i f s < - 1 , 1 < s i f -1 < s < 1 and i D = 2, i - = 1, j Q = 4 , j i - 3. Prom ( l 6 . l ) and the f a c t t h a t U*f = U*x , U?;f = ufx a a b b i = 1 , 2 , one can e a s i l y v e r i f y t h a t (16.23) [ f x ] ( b ) - [ f x ] ( a ) = U ^ x [ f ( a ) - x ( a ) ] + U 2 x [ x ' ( a ) - f ' ( a ) ] + U^x[x(b) - f ( b ) ] + U 2 x [ f ' ( b ) - x ' b ] . Then u s i n g the r e p r e s e n t a t i o n (2 .21) of f ( s ) and ( l 6 . 2 ) - ( 1 6 . 8 ) , (16.16) i n ( 1 6 . 2 3 ) , i t can be v e r i f i e d t h a t f o r x - x n (16 .24) [ f x ] ( b ) - [ f x ] ( a ) ~ 2 5 ( n + 3 ) j a [ x ( a ) ] 2 - b [ x ( b ) ] 2 j as a , b -» - 0 0 , 0 0 . 95. We now show t h a t (16.9) s a t i s f i e s the m o n o t o n i c i t y p r o p e r t y of s e c t i o n 2, i . e . t h a t the j - t h e i g e n v a l u e p.j = of (16 .17) i s not l e s s t h a n the j - t h e i g e n v a l u e X . of (16.9) J j = 0 , 1 , . . . , L e t D * be the set of a l l x e D such t h a t Lx € H and (16.25) l i m x ^ ^ ( s ) BJ + 1[X] = 0, I = 0,1 S-»+co -where B^ [ x ] - x ( 3 ) ( s ) - 2 ( s 2 + 2) x ' ( s ) , B 2 [ x ] = x " ( s ) . Then by (16.10) e v e r y e i g e n f u n c t i o n x of (16.9) i s i n D * . L e t (16.26) l x [ b = F [ ( x " ) 2 + 2 ( s 2 + 2 ) ( x ' ) 2 + ( s 4 + 4s 2 + 2 ) x 2 ] d s ; a J a 1-1 - 1 - 1 . : . We s h a l l show t h a t t h e r e e x i s t s x Q € D * such t h a t 2 f x I 1 , 2 ! _ £ ! _ = i n f 1*1 t l x J | , X € D o X 2~ and t h a t x Q i s i n f a c t the e i g e n f u n c t i o n of (16.9) c o r r e s p o n d i n g t o the s m a l l e s t e i g e n v a l u e , X Q . O b v i o u s l y L i s p o s l t i v e - b o u n d e d - b e l o w on D * s i n c e f o r any x e D * we o b t a i n by the i n t e g r a t i o n - b y - p a r t s f o r m u l a t h a t b ( L x , x ) = | x | 2 + l i m f x ( s ) B ^ [ x ] - x ' ( s ) B 2 [ x ] J a -* -00 b ^ a and hence b y (16.25) and (16.26) t h a t (16.27) ( L x , x ) = |x| 2 2 2I1XH2 • 96. We n e x t a s s e r t t h a t any i n f i n i t e s e t S c D* w h i c h i s bounded i n the | | norm has a convergent subsequence i n the L norm. L e t S be any i n f i n i t e s e t of f u n c t i o n s x € D* such t h a t 2 |x| _< C f o r some p o s i t i v e c o n s t a n t C. Then f o r any p o s i t i v e number a and any x e S J |x(s)|2ds < 4 - J s2|x(s)|2 < - i g |x|2 < C ~2 a L e t € > 0 be g i v e n and s e t a = J6c/e . Then , C e J a S i m i l a r l y (16.28) J |x(s)|2ds < a a^ 6 .-a (16.29) J _ | x ( s ) | 2 d s < | . Fo r any x € S, x ( s ) = x ( a ) + J v ( t ) d t , where v = x'. Hence by (16.26), ||v|| <_ C. S i n c e | x ( s ) - x ( r ) | = | [ ~ v ( t ) d t | _< Js^r j|v|| _< Cjs^r .s •\ r S i s e q u i c o n t i n u o u s on - a < r 3 s _< a where a = ^  ~ . A l s o S i s u n i f o r m l y bounded on t h i s i n t e r v a l , s i n c e lx(s)l < 715 + c^g ' • By A s c o l i ' s theorem t h e r e e x i s t s a u n i f o r m l y c onvergent sub-sequence {x n} on [-a,a], w h i c h i m p l i e s t h a t t h e r e e x i s t s an I n t e g e r N such t h a t +a I I V s ) - * n ( s ) | 2 d s < % - a p r o v i d e d n,m > N. But (16.28) and (16.29) a r e independent of x; 97. so for a l l n,m, J j x m - x j 2 dx < f , f j a —» Thus J txm " xnt2ds = ,lxm " xn!l ^ < e * - 0 0 provided. m,n > N. Since H i s complete there exists x e H such that l! x^ - x || -> 0, as n -» » . n By theorem 3 (\1\, pages 222-226) i t follows that the eigenfunctions x^, n = 0,1,2... of (16.9) form a system which Is complete with respect to both the f \ norm and the L norm. Further, \ x l 2 |x| 2 (lb.30) \n = —2 _ _ < — L _ f o r a l l x e D*, and (lb.31) X = — S - j j , ? for a l l x € D* such that (x,x ±) = 0, I = 0,1,2,...,n-l. Let 6 be the set of a l l x e H s a t i s f y i n g (16.25) and such that ( i ) Lx e H ( i i ) For i = 0,1,2,3,4, x ^ ^ ( s ) i s continuous on - 0 0 < s < 0 0 exqept possibly at a f i n i t e set of points; at points of discon-t i n u i t y x ^ ^ s j t ) exists (and i s f i n i t e ) for 1 = 0,1,2,3,4. By completeness of the eigenfunctions x n i t follows that (16.30) 98. and ( l 6 . 3 l ) are v a l i d f o r a l l x e G as w e l l . L e t u be the s m a l l e s t e i g e n v a l u e of (16..17) and y o 0 a c o r r e s p o n d i n g n o r m a l i z e d e i g e n f u n c t i o n . Ex tend y Q t o a f u n c t i o n y * e G, by l e t t i n g (y(s) i f a _< s _< b v o ( s ) = 1 0 " i f - » < s < a , b < s < e o . Then the I n t e g r a t i o n - b y - p a r t s f o r m u l a and (16.16) and (16.26) l e a d t o a 2 M 0 = ( L y o ^ o ) a - 8 y 0 I a + ( s 5 t y ° ( s ) ] 2 y > i n f JLSL^ = \ q x e c ||x| Thus | i Q >_ X Q . A s i m p l e p r o o f b y i n d u c t i o n shows that | i 1 >_ x ± > 1 = 1 , 2 , . . . . Theorem 5 and (16 . 2 4 ) y i e l d the f o l l o w i n g v a r i a t i o n a l f o r m u l a f o r the e i g e n v a l u e s u n = o f (16.17): U a b ~ ( 2 n + ^ + ( 2 n + 3 ) 7 r - 1 / 2 2 n + \ n i ) - 1 [ b 2 n + 1 e - b 2 - a 2 N + V A 2 ] as a , b -* -eo,oo f o r n = 0 , 1 , 2 , . . . , 17. Example 2. The f o l l o w i n g example i s d e r i v e d f r o m B e s s e l ' s e q u a t i o n and w i l l i l l u s t r a t e the m a t e r i a l i n s e c t i o n 6. Let L be the f o u r t h o r d e r d i f f e r e n t i a l o p e r a t o r d e f i n e d on the h a l f - o p e n i n t e r v a l (0 ,1 ] by Lx = L 0 L 0 x where 99. s n f i x e d , n > 3. Then ( 1 7 . 1 ) Lx = iT[BX-]" -[(^gi-l) * ' ] ' + ( S ^ 2 2 ) x' We have P 0 ( s ) = s, k(s) = s and from (0.4) 2 (17.2) [uv] =• 2 n s + 1 (uy' - u'v) + (u"v - uv" ) +s(u^^v -- u"v' + u ' v "- u v P ) ) . Let D(0,l] be the set of a l l x e H(0,l] such that 3 (3) ; (1) x 6 C (0,1] and x ^ > ; i s a b s o l u t e l y continuous on every closed s u b i n t e r v a l of (0,1]; and ( i i ) ( 1 7 . 3 ) x ( l ) =' x ' ( l ) = 0. Then the b a s i c problem i s the eigenvalue problem (17.4) , Lx = \x , x e D(0,1]. The d i f f e r e n t i a l equation Lx = 0 has the f o l l o w i n g l i n e a r l y independent s o l u t i o n s : (17.5) «p1(s) = s~n ; <p5(s) = s n ; / \ -n+2 , v n+2 <p2(s) = s ; cp^(s) = s From ( 1 7 . 3 ) i t f o l l o w s e a s i l y that 0 i s not an eigenvalue of (17.4); Hence we choose iQ = 0 and cp^, i = 1,2 , 3,4 as the b a s i c s o l u t i o n s . By c o n s t r u c t i n g the Green's f u n c t i o n f o r L (using (17.5) associated w i t h the boundary c o n d i t i o n s ( 1 7 . 3 ) , one can e a s i l y 100. v e r i f y t h a t (17.4) has a countable set of r e a l eigenvalues and a co r r e s p o n d i n g s et of orthonormal e i g e n f u n c t i o n s complete i n H(0,1] . Since 0 i s a r e g u l a r s i n g u l a r i t y f o r L, one ob t a i n s by c o n s i d e r a t i o n of the i n d i c i a l e q uation {[3], pp. 122-127) t h a t each e i g e n f u n c t i o n x of (17 .4) s a t i s f i e s (17.6) l i m x ^ ^ s ) = 0 , i = 0,1,2; s - 0-l i m x ^ ( s ) = C . s - 0 Then s i n c e 1 2 i^- 2 ( L x , x ) l = f { s [ x " ] 2 + ( 2 n s + ^ [ x ' ] 2 + ( n j 4 n ) x 2 ] d s + l i m ^( s x " ) ' x - sx"x' - ( 2 n s + 1 ) x x ' j l  2? a - 0 by the' I n t e g r a t i o n - b y - p a r t s formula one ob t a i n s by (17 .3) and.(17.6) t h a t f o r any e i g e n f u n c t i o n x of ( 1 7 . 4 ) , ( L x , * ) 0 > ( H x i l o ) 2 and hence t h a t a l l eigenvalues of (17.4) are p o s i t i v e . Let x be any eigenvalue of (17.4) and l e t I = X"^ 4, I > 0. Then any cor r e s p o n d i n g e i g e n f u n c t i o n x has the form x ( s ) = A Jn(ls) + B J n ( U s ) where A,B are constants and J" n denotes a B e s s e l f u n c t i o n of the f i r s t k ind. By (17.3) one can e a s i l y deduce t h a t f o r the eigenvalue \ there e x i s t s e x a c t l y one l i n e a r l y independent e i g e n f u n c t i o n x. Hence a l l eigenvalues of (17.4) have m u l t i -p l i c i t y 1. 1 0 1 . For 0 < a < 1 , l e t D[ a , l ] denote the set of a l l y € H[a,l] such that ( i ) y e C^[a,l] and y ^ ^ i s a b s o l u t e l y continuous on [ a , 1 ] ; ( i i ) Ly € H[a,l]j and ( i i i ) y(a) = yt(a) = y ( l ) = y ' ( l ) = 0 . Then the perturbed problem i s the r e g u l a r s e l f - a d j o i n t eigen-value problem ( 1 7 . 7 ) Ly = uy , y e D [ a , l ] , Let X j be the j - t h eigenvalue of ( 1 7 . 4 ) , X 1 < X 2 < ••• < ^ j < •••• T i i e corresponding eigenf unction x j ( s ) can be expressed i n the form: ( 1 7 . 8 ) x.(a) = i 5 n C j [ J n ( U 0 ) J n ( ^ j s ) - Jn(l^)J(±l^)] 1/4 where C. i s the n o r m a l i z a t i o n constant and l. = X/ , I . > 0 . J J J J By i n s p e c t i n g the s e r i e s r e p r e s e n t a t i o n of J n ( s ) one obtains that ( 1 7 . 9 ) . x j ( a ) ~ U ^ l t j ) - i n J n ( * j ) ! - n I 1 • ^ xL(a) ~ na"" 1x.(a) ; J J — 2 x".(a) ~ n(n - l ) a " x . ( a ) ; J J x ? ) ( a ) ~ n(n - l ) ( n - 2)a"5x1(a) as a - 0 . I t i s e a s i l y v e r i f i e d that c o n d i t i o n s ( 2 . 7 ) , ( 2 . 9 ) , ( 3 - 3 ) , ( 3 . 5 ) and ( 3 . 6 ) are s a t i s f i e d i f we choose h(s) = 1 , 102. i ^ = 2 and j Q = 3. In p a r t i c u l a r , -0 ~ Ca11 as a -* 0. By actual c a l c u l a t i o n one may obtain-that f (a) ~ x (a ) ; f " ( a ) ~ - 3 n(n - l ) a ~ 2 x ( a ) ; f (a) ~ n a ^ x C a ) ; f ® ( a ) ~ n (n - l ) ( 5 n + 2)a" 5x(a) as a - 0. Hence by (17.2) and (17.9) (17.10) [fx](a) ~ 8 n 2(n - 1 )a" 2[x(a) j 2 - . Also by. (17.2) and (17 .3) and the fac t that • :' N • f ( l ) = f'(-i') = x ( l ) = x'(1) = 0 one obtains (7.11) [ f x ] ( l ) = 0 . By consideration of the biharmonic operator (iterated Laplacian) L = AA (See [12]) on the unit disc one can deduce that (17.4) s a t i s f i e s the monotonicity property of section 6. Then theorem 9 and (17-9) - .(17.11) y i e l d the v a r i a t i o n a l formula 4 ~ X, + 8 n 2 ( n - l ) a - 2 r x j ( a ) ] 2 . . as a 0, where C. i s the normalization constant defined by ( 1 7 . 8 ) . J It i s e a s i l y seen that the remaining terms on the right of ( 6 . 8 ) , namely fro " 0 ( f , f £ , ^ 1 ) ^ 0 ( 1 ) are of smaller asymptotic order for this.example and hence may be disregarded. 103. 18. Example 3. As an example o f the t h e o r y i n s e c t i o n 8 c o n s i d e r the o p e r a t o r L d e f i n e d on the h a l f - o p e n i n t e r v a l ( 0 ,1 ] hy (17 .1) w i t h n = 0. Then (18.1) Lx = i { ( s x " ) " - ( i x ' ) ' ] . L e t cp^, I = 1,2,3,4 denote the f o l l o w i n g l i n e a r l y independent s o l u t i o n s of Lx = 0: (18.2) cp 1 (s) = l o g s ; cp 2 (s) = 1 ; cp^(s) = s l o g s ; cp 4 (s) = s . Then f o r i = 1,2,3,4, cp i 6 H(0 , l ] where the i n t e g r a l s r e p r e -s e n t i n g the i n n e r p r o d u c t and norm f o r H(0 , l ] are t a k e n w i t h r e s p e c t t o the w e i g h t f u n c t i o n k ( s ) = s. S e t t i n g n = 0 i n (17. 2) .we o b t a i n (18.3) [ u v ] ( s ) = i ( u v ' - u v ' ) + (u"v - u v " ) + s ( u ^ ^ v - u " v ' + u ' v " - u v ^ ) . By G r e e n ' s f o r m u l a (0.5), [cp^cpjj(s) i s c o n s t a n t on 0 < s _< 1, i , j = 1,2,3,4, and by a c t u a l c a l c u l a t i o n (18.4) 0 0 4 4 0 0 -4 0 -4 4 0 0 -4 0 0 0 L e t X 1 , i = 1 ,2,3 ,4 , be d e f i n e d b y : 104. (18.5) X l ( s ) = cp5(s) ; X 2 ( s ) = cp^(s) ; X 5 ( B ) = {(P x(s) + cp2(s) + cp^(s) - cp^(s)] ; X j ^ s ) = {•2cp1(s) + <p2(s) - e p 4 ( s ) ] . T h e n , w i t h t h e h e l p o f ( l8 .4) . , I t i s e a s i l y v e r i f i e d t h a t [ x ± X j ] ( 0 ) = 0 , i , j = 1,2 ; [ x i X j ] ( D = 0 , - i , J = 3,4 . F r o m (18.4) i t a l s o f o l l o w s t h a t (18.6) [ep 1x j](0) £ 0 , i , J = 1,2, i { J . ; [cpjXjHl) / 0 , i , j = 3 ,4 , i ft j . L e t D(0,1] b e t h e s e t o f a l l x e H (0,1] s u c h t h a t ( i ) x e C ^ ( 0 , l ] a n d x ^ ^ i s a b s o l u t e l y c o n t i n u o u s o n c l o s e d s u b i n t e r v a l s o f ( 0 , 1 ] ; ( i i ) x s a t i s f i e s t h e e n d c o n d i t i o n s (18.7) [xxj ] ( 0 ) = 0 , J = 1,2, [ x X j K D = 0 , j = 3,4 . ( N o t e t h a t t h e a b o v e e n d c o n d i t i o n s a t s = 1 a r e x ( l ) = x ' ( l ) " = 0.) T h e b a s i c p r o b l e m ( c o r r e s p o n d i n g t o (8.1)) i s (18.8) Lx = \x , x e D(0,1], A n ' I m m e d i a t e c o n s e q u e n c e o f ( l 8 . 6 ) i s t h a t X = 0 I s n o t a n e i g e n v a l u e o f ( l 8 . 8 ) . H e n c e w e c h o o s e lQ = 0 a n d cp i 5 I = 1 ,2 ,3,4, t o b e t h e b a s i c s o l u t i o n s . B y c o n s t r u c t i n g t h e G r e e n ' s f u n c t i o n 105. f o r L , ( u s i n g ( l 8 . 2 ) ) a s s o c i a t e d w i t h the boundary c o n d i t i o n s ( l 8 . 7 ) , one can e a s i l y v e r i f y t h a t (18.8) has a c o u n t a b l e se t of r e a l e i g e n v a l u e s a c c u m u l a t i n g o n l y a t » and a c o r r e s p o n d i n g s e t o f o r t h o n o r m a l e i g e n f u n c t i o n s complete i n H ( 0 , l j . By c o n s i d e r a t i o n of the i n d i c i a l e q u a t i o n f o r Lx = Xx one o b t a i n s t h a t each e i g e n f u n c t i o n of (18.8) i s such t h a t (18.9) x ^ ^ s ) = 0 ( s 2 " i l o g s ) , I = 0 ,1 ,2 ,3 . Then s i n c e (Lx,x)J = P { s [ x " ] 2 + -|tx']2}<Js * a ^ 0 { ( « " ) ' « - " " » • - ^ l ' 1 a by the I n t e g r a t i o n - b y - p a r t s f o r m u l a , one o b t a i n s by (18.7) and (18.9) t h a t f o r any n o r m a l i z e d e i g e n f u n c t i o n ' x of (18.8) , ( L x , x ) > 0. T h i s i m p l i e s t h a t the e i g e n v a l u e s of (18.8) are a l l p o s i t i v e . L e t X be any e i g e n v a l u e of (18.8) and l e t X = k 1 / 4 , k > 0. Then l i n e a r l y i n d e p e n d e n t s o l u t i o n s of Lx = xx are J Q ( k s ) , J Q ( i k s ) , Y 0 ( k s ) and Y Q ( i k s ) where J Q and Y Q are B e s s e l f u n c t i o n s of the f i r s t and second k i n d r e s p e c t i v e l y . By i n -s p e c t i n g the s e r i e s r e p r e s e n t a t i o n s ' o f J Q ( s ) and Y Q ( s ) (See [15]) one o b t a i n s t h a t the n o r m a l i z e d e i g e n f u n c t i o n s x c o r r e s p o n d i n g 4 t o X = k must be of the f o r m (18.10) x ( s ) = C x { A j Y 0 ( i k s ) - Y D ( k s ) ] + B x [ J 0 ( i k s ) - J 0 ( k s ) ] } , where A.. , B. , C. are c o n s t a n t s g i v e n by A. A. A. i o 6 . A x = J Q ( i k ) - J Q ( k ) ; B. = Y Q ( k ) - Y 0 ( i k ) ; A x [ Y Q ( i k s ) - Y 0 ( k s ) ] + B x [ J G ( i k s ) - J Q ( k s ) ] Then i t f o l l o w s e a s i l y f r o m (18 . IO) t h a t each e i g e n v a l u e of (18.8) has m u l t i p l i c i t y 1. -1 F o r 0 < a < 1, l e t D [ a , l ] denote the set of a l l y e H [ a , l ] such t h a t ( i ) y e C ^ [ a , l ] and y ^ i s a b s o l u t e l y c o n t i n u o u s on [ a , l ] ; ( i i ) Ly e H [ a , l ] j and ( i i i ) y ( a ) = y ' ( a ) = y ( l ) = y ' ( l ) = 0 . Then the p e r t u r b e d p r o b l e m i s the r e g u l a r s e l f - a d j o i n t e i g e n -v a l u e p r o b l e m ( l 8 . l l ) Ly = uy , y e D [ a , l ] . I t i s e a s i l y v e r i f i e d t h a t c o n d i t i o n s ( 2 . 7 ) , ( 2 . 9 ) , ( 3 - 3 ) , (3-5) and (3.6) are s a t i s f i e d i f we choose i ^ = 2, j Q = 3 and h ( s ) -1 I log s| if 0 < s < e i f e " 1 < s < 1 . C o n d i t i o n (7.15) i s a l s o s a t i s f i e d w i t h p a ~ C a 2 ( l o g a ) 2 = o ( l ) as a -» 0. By ( l8.10) one o b t a i n s t h a t 107. (18.12) x'(a) ~ lx(a) ; x"(a) ~ -^x(a) 1 a 2 , ~ •* x( a^log a a) as a -» 0 . By actual c a l c u l a t i o n one obtains: (18.13) f ( a ) ~ x(a) ; f"(a) ~ 2 x(a) ; a, f t ( a ) ~ | x(a) ; f ^ ( a ) - x(a) ST as a -• 0. Hence ( 1 8 . 3 ) , ( 1 8 . 1 2 ) , and (18.13) y i e l d (18.14) [fx](a) ~ 8 a " 2 [ x ( a ) ] 2 , a - 0 . Also since f ( l ) = f ' ( l ) = x ( l ) = x ' ( l ) = 0, ( l 8 . 3 ) implies that (18.15) [ f x ] ( l ) = 0. Since a monotonicity property i s known to hold ( i n the same way as i n example 2 ) , theorem 15 together with ( 1 8 . 1 0 ) , (18.14) and (18.15) y i e l d the following v a r i a t i o n a l formula for the eigenvalues n a of ( l 8 . l l ) , i = 1 , 2 , . . . : ^a - x i ~ * * W x ± * 2 ( l o e a ) 2 as a -» 0, i = 1 , 2 , . . . . Note that the remaining terms on the ri g h t of ( 8 . 2 ) are a l l 0[a^(log a)^] as a -» 0 and hence may be disregarded. 108. BIBLIOGRAPHY [ l ] A . C . A i t k e n , " D e t e r m i n a n t s and m a t r i c e s " , I n t e r s c i e n c e P u b l i s h e r s I n c . , New Y o r k , 1956. [2] E . A . C o d d i n g t o n , "The s p e c t r a l r e p r e s e n t a t i o n of o r d i n a r y s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r s " , A n n a l s of Math. 60 (1954) p p . 192-211. [3] E . A . C o d d i n g t o n and N. L e v i n s o n , " T h e o r y of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s " , M c G r a w - H i l l , 1955. [4] R. Courant and D. H i l b e r t , "Methods of m a t h e m a t i c a l p h y s i c s I " , I n t e r s c i e n c e P u b l i s h e r s I n c . , New Y o r k , 1963. [5] N. D u n f o r d and J . T . Schwar tz , " L i n e a r O p e r a t o r s , P a r t I I : S p e c t r a l t h e o r y " , I n t e r s c i e n c e P u b l i s h e r s I n c . , New Y o r k , 1963. [6] K. K o d a i r a , "On o r d i n a r y d i f f e r e n t i a l e q u a t i o n s of any even  o r d e r and the c o r r e s p o n d i n g e i g e n f u n c t i o n e x p a n s i o n s " , Amer. J . Math. 72 (1950)pp. 502-544. [7] S . G . M i k h l i n , " V a r i a t i o n a l methods i n m a t h e m a t i c a l p h y s i c s " , The M a c M i l l a n C o . , New Y o r k , 1964. [8] M . A . Neumark, " L i n e a r e D i f f e r e n t i a l o p e r a t o r e n " , Akademie-V e r l a g , B e r l i n , i960. [9] F . R e l l i c h , " S p e c t r a l t h e o r y of a second o r d e r o r d i n a r y  d i f f e r e n t i a l o p e r a t o r " , New York U n i v e r s i t y , 1953. [10] F . R i e s z and B. S z - N a g y , " F u n c t i o n a l a n a l y s i s " , B l a c k i e and Son, 1956. [11] C . A . Swanson, " A s y m p t o t i c e s t i m a t e s f o r l i m i t c i r c l e p r o b l e m s " , P a c i f i c J . M a t h . , 11 (1961) p p . 1549-1559. ' 109. [12] C . A . Swanson, "Domain p e r t u r b a t i o n s of the Dinarmonic o p e r a t o r " , Can. J . M a t h . , 17 (1965) p p . 1053-10b3. [13] C . A . Swanson, " A s y m p t o t i c e s t i m a t e s f o r l i m i t p o i n t p r o b l e m s " , P a c i f i c J . M a t h . , 13 (1963) p p . 305-316. [14] A . E . T a y l o r , " I n t r o d u c t i o n t o f u n c t i o n a l a n a l y s i s " , John W i l e y and Sons , I n c . , New Y o r k , 1964. [15] E . T . W h i t t a k e r and G . N . Watson, " A course of modern a n a l y s i s " , Cambridge U n i v e r s i t y P r e s s , 1952. 

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