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_.»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 = ~* 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, . . . ,+ ' 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(4_(b) ( 2 . 3 ) S a(i*d) = o a ( i , J ) U C P . ^ ; 5b(i»J) = a b ( i , j )ll < ( 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 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«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 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. + 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 - 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 >&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 _,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 < 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 | ,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] +) = 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 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) 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 ) 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 _ ), i = 1,2, . . . , n ; j = 1,2, . . . , m A. . = 1 J 1 [_,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 ; ( 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) + 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.