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Properties of eigenvalues of singular second order elliptic operators Welsh, K. Wayne 1970

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PROPERTIES OP EIGENVALUES OF SINGULAR SECOND ORDER ELLIPTIC OPERATORS by K. WAYNE WELSH B.Sc. University of Alberta, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. i n the Department of MATHEMATICS . We accept this' thesis as conforming to the required standard The University of B r i t i s h Columbia March 1970 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i ] a b l e f o r reference and study, I f u r t h e r agree t h a t p e r m i s s i o n - f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or 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 in s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f _ _ 4 ? f i 4 t e e _ 1 _ r f c 4 e _ _ _ The U n i v e r s i t y of B r i t i s h Columbia • Vancouver 8 , Canada Date -Sad-Ap-Al-- 1970-Supervisor: Colin ¥. Clark i i . ABSTRACT This thesis investigates the properties of the L^-eigenvalues of singular, e l l i p t i c , second order operators, p r i m a r i l y the operator L defined by ^(L) = wJ j 2(G,V) n {u e L 2(G) | -Au + Vu e L 2(G)} Lu = -Au + Vu f o r u e -#(L) . Here the "potential function", V(x)., i s such that .(I! -.||^ 2 + J J-| 2V(x)dx)* i s a norm rai C-(G) ( I! • l [ ^ 2 bein the usual norm i n the Sobolev space ¥ l j 2 ( G ) ) and Wlj,2(G,V) i s the completion of C Q ( G ) i n the metric from this norm, i d e n t i f i e d with a subset L 2(G) ; A i s the Laplacian and G i s an a r b i t r a r y open domain of E n . Several s u f f i c i e n t conditions are given on V and on G i n order that L have spectrum s a t i s f y i n g (-oo,a) fl a(L) c: a (L) , for some re a l number a (CT(L), a (L) denote the spectrum and point spectrum of L , r e s p e c t i v e l y ) . The properties of these lower eigenvalues are invest-igated by examining the eigenvalues of a coercive b i l i n e a r form corresponding to the operator. Such a form B , having domain J 5 - , say," i s defined to have eigenvalue \ e $ -with corresponding eigenfunction u e . f r i f B[u,f] = \(u,f) f o r i i i . a l l f e i> . V a r i a t i o n a l properties are discussed i n d e t a i l ; In p a r t i c u l a r , a condition i s given which ensures that the numbers, sup i n f B[u,u] (the sup and i n f being over approp-r i a t e sets involving i> and n ) are eigenvalues of B . These properties are applied to L to generalize the well-known c l a s s i c a l property '(G bounded) of monotonic dependence of the eigenvalues on the underlying domain G : G cz G* implies X n _ \* for corresponding eigenvalues, with s t r i c t i n c l u s i o n implying s t r i c t inequality. A few m i s c e l l -aneous properties of the eigenvalues and eigenfunctions then follow from this dependence. i v . TABLE OP CONTENTS page INTRODUCTION 1 CHAPTER ONE - Examples of Semi-Discrete Operators 1 6 A. Definitions 6 B. Examples 7 ( i ) Priedrichs Extension 7 ( i i ) The Laplacian Operator 8 ( i i i ) The Schrodinger Operator 9 (iv) General Second Order 2 5 CHAPTER T¥ 0 - Va r i a t i o n a l Properties 2 6 A. V a r i a t i o n a l Properties f o r Semi-Discrete Operators 2 6 B. V a r i a t i o n a l Properties for B i l i n e a r Forms 2 9 CHAPTER THREE - Dependence of Eigenvalues on the Domain 4 8 A. Weak Monotonicity Property f o r Forms 48 B. Strong Monotonicity Property f o r Forms 5 2 C. Summary 5 5 CHAPTER FOUR - Applications of Monotonicity 5 7 A. Symmetry . / 5 7 B. Miscellaneous 6 0 BIBLIOGRAPHY . 6 2 V . ACKNOWLEDGEMENT The author would l i k e to thank his supervisor, Dr. Colin W. Clark, f o r suggesting the topic of this thesis and for the encouragement and helpful advice given during i t s preparation. He would also l i k e to extend his appreciation to Dr. R. A. Adams and Dr. A. T. Bui for t h e i r c a r e f u l reading and constructive c r i t i c i s m s of the draft form of this thesis. The generous f i n a n c i a l support of the H.R. MacMillan family and of the National Research Council of Canada i s also g r a t e f u l l y acknowledged. INTRODUCTION For eigenvalue problems, the Lg theory of d i f f e r -e n t i a l equations (cf. [11], [23], [25'}, [26], [ f ] ) asks f o r complex numbers X and functions u e L 2(G) for which Lu = X.u i n L 2(G) , where L i s some operator i n Lg(G) , usually a generalized d i f f e r e n t i a l operator. The purpose of this thesis i s to study properties of such eigenvalues i n connection with operators L that correspond to singular second order d i f f e r e n t i a l operators - the s i n g u l a r i t i e s a r i s i n g from the unboundedness of the underlying space, G , of the domain of L . .  Before o u t l i n i n g the r e s u l t s , we l i s t some termin-ology, as well as some of the basic d e f i n i t i o n s of L^-theory. References to the_ bibliography w i l l be given by a bracketed number, occasionally with the author's name included. Let § = (§^,...,5 ) be a point i n E n , Euclidean n-space. Following the notation of Schwartz, denote i% = .. . ? , f o r a multi-index a = (a n ,. .. ,a ) ' of I n * v I3 3 nJ a a i % ' integers a i . Also denote D = D^ ... D n , where D. = , and I a I = sx, +. .. +a l ox^ 3 1 1 1 n B R(0) or B R w i l l denote the b a l l of radius R , centered at 0 ; B R = {x e" E n | [x| <_ R} . ¥e always use G - G^  for the set theoretic difference [x | x e G and x | G 1) . For an open set G c= F / 1 , c"(G) w i l l denote the space of a l l i n f i n i t e l y d i f f e r e n t i a b l e complex-valued functions having compact support i n G (test functions); C m(G) w i l l be the set of m-times Continuously d i f f e r e n t i a b l e functions i n G . The inner product and norm i n Lv^G-^) for any G l _ G w i l l be denoted (U,v) G = J u ^ d X ; || U ||2 fl. = I |u|2dx , °9*1 :GX °> ul ' G with the subscripts G^  omitted when G^  = G and the o omitted when no confusion w i l l a r i s e . For non-negative integers m,p and u e C m(G) , l e t H * US n - , ? I (D au| Pdx m ' p " a |<m "G F i n a l l y , we require the following Sobolev spaces: ¥ m / p ( G ) = completion of {u e C m(G) | || u || < «} m, p with norm || • || ; W^ P(G) = closure of cf (G) i n W^^G) . Elements of ¥ m* p(G) can be represented another way using weak ( d i s t r i b u t i o n , generalized L„) d i f f e r e n t i a t i o n : 3. D e f i n i t i o n I f u e L ^ 0 0 (G) , then u has weak a-derivatlve D au i f there exists v e L 1 0 0 (G) such that J f vdx = ( - l ) ' a ' J in D a f dx for any f e C~(G) . We write v = D u . Using t h i s d e f i n i t i o n , W ^ G ) = [u e L P(G) | D au e L P(G) ; |a| £ m} with norm || • || m, p (cf. Meyers, Serrin [24-]). The above spaces can be defined f o r r e a l valued functions and r e a l scalars. I f such a r e s t r i c t i o n i s required, i t w i l l be e x p l i c i t l y stated (for instance, i n Theorem 2.4); s i m i l a r l y the c o e f f i c i e n t s of operators or forms w i l l be assumed complex valued unless otherwise stated. In Chapter One we discuss, primarily, two singular operators - the generalized Laplacian and generalized Schrodinger operators: l e t Lu = -Au + Vu f o r u e C*(G) ; ' under various conditions on the p o t e n t i a l V(x) , and on the underlying space, G , there Is a s e l f - a d j o i n t extension for L (Friedrich'-s extension) which i s c a l l e d the generalized Schrodinger operator (Laplacian, i f V . = 0). The domain of this extension i s characterized and the question of existence of i s o l a t e d eigenvalues i n the lower part of the spectrum i s investigated under various additional r e s t r i c t i o n s on V and G . 4. In some cases the spectrum turns out to be discrete; i n others, an upper bound i s given for the discrete part of the spectrum. Operators having only i s o l a t e d eigenvalues i n the lower part of the spectrum are c a l l e d semi-discrete, and we are concerned with properties of the lower eigenvalues only. For the . Schrodinger operator, the basic condition given governing the behavior of V at i n f i n i t y i s due to Clark [8]. We also mention a condition due to Persson [28], stronger than the above one, which was used to investigate semi-discreteness of L . In general our results require weaker conditions f o r the boundary of G to s a t i s f y than those given by either Clark or Persson. In Chapter Two the v a r i a t i o n a l (max-min) aspects of the eigenvalues of semi-discrete operators are discussed. In 1 r p a r t i c u l a r , a converse to the usual theorem i s given: when are the numbers X n = sup inf,(Lu,u) (the supremum and infimum being over appropriate sets involving n and the domain of L ) , eigenvalues of L ? After giving a s u f f i c i e n t condition, i t i s shown that A. i s an eigenvalue of L with corresponding eigenfunction u i f and only i f there i s a sesqui-linear (hereafter c a l l e d b i l i n e a r ) form, B[*,-] , associated with L , s a t i s f y i n g B[u,f] = \(u,f) for a l l f e C*(G) . Such an equation serves as a d e f i n i t i o n f o r eigenvalues and eigen-functions of any a r b i t r a r y form, and the' same question as above, concerning existence of eigenvalues, i s investigated. A similar s u f f i c i e n t condition i s given; some of the proofs here (Lemmas 2.6, 2.7) require a r e s t r i c t i o n to the f i e l d of reals. Hestenes [18] has investigated the above problem for compact operators and Berkowitz [6] for se l f - a d j o i n t operators. These v a r i a t i o n a l properties are used i n Chapter Three f o r discussing the dependence of the eigenvalues of b i l i n e a r forms on th e i r underlying domain G ; the c l a s s i c a l property, that corresponding eigenvalues are non-decreasing as G shrinks, remains v a l i d f or forms under quite weak conditions. The stronger property, that the corresponding eigenvalues increase s t r i c t l y as G shrinks, i s v a l i d f or the lower eigenvalues of forms that correspond to e l l i p t i c operators which s a t i s f y a unique continuation p r i n c i p l e l o c a l l y . I t i s also v a l i d f o r some of the semi-discrete operators discussed i n Chapter One; . for example, i f L = -A + V" has V(x) -* » as | x | -» » 9 x e G then the eigenvalues of L increase s t r i c t l y as G shrinks. This i s the main property discussed, and appears to give new res u l t s f o r some of the singular operators, as the one mentioned above. The l a s t chapter uses this strong monotonicity r e s u l t to prove a symmetry resu l t f o r the generalized Laplacian operator (G unbounded) which i s well-known f o r the c l a s s i c a l operator: . a symmetry of G induces a symmetry i n the eigenfunctions. A few miscellaneous applications complete the chapter. The theorems and lemmas are l a b e l l e d by order within a chapter; f o r example, Theorem J5..1 refers to the f i r s t theorem of Chapter Three. CHAPTER ONE SEMI-DISCRETE OPERATORS A. Definitions Let H be a Hilbert space and L be an operator with domain -#(L) £E H . Following the usual d e f i n i t i o n s , [14], we say L i s symmetric i f (Lu,v) = (u,Lv) fo r any u,v e .fr(L) > I f ^(L) i s dense i n H , then the (Hi l b e r t -space) adjoint of L i s the operator L* having domain £(L*) = {u e H | 3 u* e H with (Lv,u) = (v,u*) for any v e i^L)} and defined by L*u = u* for u e £(L*) . If L = L* , then L Is said to be s e l f - a d j o i n t . We s h a l l be interested i n operators of the following type: D e f i n i t i o n A s e l f - a d j o i n t l i n e a r operator L w i l l be c a l l e d semi-discrete, i f there exists a rea l number a such that ( - 0 0 , a ) fl c(L) c a (L) , where a(L) , a (L) denote the spectrum ir P and point spectrum of L respectively (cf. [30]). 7 . B. Examples ( i ) Friedrich's Extension. For bounded operators, the well-known Riesz-Schauder theory, [14], shows that a l l s e l f - a d j o i n t compact operators are semi-discrete. For unbounded operators, we r e s t r i c t our 'attention to s e l f - a d j o i n t second order e l l i p t i c operators ( i f L = E a (x)D a , then L i s e l l i p t i c i n G i f and only i f |a|<2 a for any x e G and any r e a l 5 = (5^,...,§ n) ^ 0 , we have S a (x) e / 0 ; cf. [53). |tt|=2 a Under various conditions on the region G , and on the "potential", V(x) , the Laplacian operator, -A , and the Schrodinger operator, -A + V , defined on . C*(G) , have s e l f -adjoint extensions which are semi-discrete. In order to be e x p l i c i t about the domains of the extensions, we f i r s t make some remarks concerning the construction of the Fr i e d r i c h ' s extension (hereafter abbreviated to F.E.) of a symmetric operator,, cf. [30]; then we give conditions such that the F.E.'s of the above operators are semi-discrete. In some cases the conditions w i l l imply that the operators have compact resolvents. I f Ir- i s a symmetric operator with domain, -5"(L) , dense i n a Hil b e r t space H , and i f L i s semi-bounded, then there i s a "minimal" s e l f - a d j o i n t extension which can be 8. associated with L , ca l l e d the F.E. of L . Since we are only-interested i n eigenvalue problems, the addition of constants to the operator, or multiples of the operator, w i l l not a f f e c t the general theory, so that we may take semi-bounded to mean (L f , f ) >. ( f , f ) for any f e £(L) . With L we associate any b i l i n e a r form B [ - , - ] f o r which B[f,g] = (Lf,g) f o r f,g e ^ (L) . Since B [ f , f ] >_ ( f , f ) on £(1>) x ^ (L) , B becomes an inner product on ^(L) . The abstract completion of -#(L) i n the metric from this inner product may be i d e n t i f i e d with a subset, H , of H i n the usual way. F i n a l l y , l e t B ^ be the extension by continuity of. B to H Q x H q . Then the F.E. of L ', L± , i s defined as the (unique) s e l f - a d j o i n t extension of L with ^'(L 1) (dense) i n " H Q and with ( i ^ f ^ g ) = B 1 [ f , g ] f o r any' f € ^-(L-^) , g £• H . In the construction i t i s also seen that L-^ exists as a bounded operator from H to -5"(L^ ) . ( i i ) The Laplacian Operator. Let Lu = -Au + u on C*(G) <= L g(G) = H , and B[u,v] = (vu,vv) + (u,v) . Symmetry and semi-boundedness are obvious on c"(G) • Hence B - ^ [ - , - ] i s defined on W^ 2(G) x W^ 2(G) and the F.E. of L i s L± = -A + I where derivatives are i n the weak sense, and ^ ( L 1 ) = W^ 2(G) 0 (u e L 2(G) | Au e L 2 ( G ) j . The proofs of these facts are e s s e n t i a l l y the same as f o r the next example, where they are done i n d e t a i l . We c a l l L-, the Laplacian. Standard usage would c a l l -A the Laplacian, hut we carry along the I ( i n the next example also) f o r ease of notation l a t e r . ( i i i ) The Schrodinger Operator. Let Lu = -Au + (V + I)u on C*(G) cr L g(G) = H . Clearly some conditions are required on V so that L i s an operator i n Lg(G) . For this we w i l l always assume that V i s square integrable on bounded sets of G , since this and even stronger conditions w i l l be needed l a t e r . We w i l l also always consider V to be a r e a l valued measurable function, so that L i s symmetric. In order to obtain semi-boundedness, the following conditions on V turn out to be s u f f i c i e n t : CONDITION A ( i ) For any compact set F c E n , there exists a constant K , depending only on the diameter of F , such that J V|f [ 2dx >_ K J |f | 2dx f o r any f e L 2(G) ; ( i i ) FflG FflG • C f x ) - 2 l i m i n f —'- v V(x)] = u > -» } where C(x) i s the | x |-»°=,xeG c length of the largest l i n e segment i n G containing x and p a r a l l e l to the x^-axis, and c i s a c e r t a i n constant appear-ing i n Lemma 1 .1 below. If V were bounded on bounded sets, then ( i ) would be true. U t i l i z i n g an unpublished proof due to R. D. Moyer (Lemma l . l ) we are able to show semi-boundedness. 10. Lemma 1 . 1 F o r C(x) d e f i n e d as i n C o n d i t i o n A, t h e r e i s a c o n s t a n t c > 0 s u c h t h a t . J C ( x ) - 2 m | f ( x ) | 2 d x < c S J | D a f ( x ) | 2 d x G a =m G f o r any f e w^> 2(G) , m = 1,2,.. P r o o f : L e t ( ¥ _ ( x ) , ¥ + ( x ) ) d e n o t e t h e l a r g e s t open l i n e s e g -ment i n G c o n t a i n i n g x and p a r a l l e l t o t h e x - a x i s , and l e t *n = ( x l v " J V l ^ ' ^ s u ^ ^ i c e s t o f i n d a c o n s t a n t c s u c h t h a t t h e i n e q u a l i t y h o l d s f o r any f e C~(G) , so s e t s u p p o r t f = K . F o r x € K , T a y l o r ' s f o r m u l a y i e l d s ^m-1 ¥ (x) ( m " l ) ! . x ( x - y ^ .m f (x) = F -7^-7- V f ( ^ , y ) d y n B u t t h e n t h e C a u c h y - S c h w a r z i n e q u a l i t y g i v e s T T 7 „ \ ( m - l ) . l ~ / \ dx ¥ ( x ) v y ¥ (x) n However on any component I o f G fl {x | x^ = c o n s t a n t } , C(x) i s c o n s t a n t , so t h a t V x ) m P C ( x ) - 2 m | f ( x ) | 2 d x r i < f d x n J | A _ f ( x ; , y ) | 2 d y 1 ' C ( x ) 1 ¥ (x) 3 x n ¥ f x ) + m < c J - ' | ^ f ( x ; , y ) | 2 d y ¥ (x) 3 x n S i n c e K fl {x | = c o n s t a n t } i s c o n t a i n e d i n a f i n i t e number o f s u c h components, we i n t e g r a t e o v e r t h e i r u n i o n t o o b t a i n 1 1 . J C ( x ) - 2 m | f ( x ) | 2 d x n < c J l - i - - f ( x ) | 2 d x n - 0 3 -CO S X j ^ and f i n a l l y J C ( x ) - 2 m | f ( x ) | ^ d x < c Z f | D a f ( x ) | ^ d x G |a|=m J G • We c a n now show t h a t C o n d i t i o n A i m p l i e s t h a t L i s bounded b e l o w . By A ( i i ) , t h e r e i s an R > 0 w i t h + V ( x ) > N f o r some N > -» , f o r a l l x e G , c |x| _ R . U s i n g Lemma 1 . 1 , w i t h m = 1 , we have f o r any f € C-(G) ( L f , f ) = ,f [ | v f [ 2 + (V(x) + l ) | f | 2 ] d x G >. p [C (x) + v ( xy + u | f | 2 d x G c T h e r e f o r e ( L f , f ) >_ J* N | f | 2 d x + J V ( x ) | f | 2 d x s i n c e G-B R GHB R " ' £(x ) , c > 0 . By A ( i ) , t h e s e c o n d t e r m i s bounded b e l o w , so t h e r e e x i s t s a c o n s t a n t N-^  w i t h ( L f , f ) >_ N-JI f || f o r any f e c"(G) . I n l i n e w i t h o u r e a r l i e r comments a b o u t c o n s t a n t s , we assume N-^  = 1 . 12. Remarks. 1. We note that the above argument, and Lemma 1.1 as well, would be v a l i d i f A ( i i ) were stated i n terms of [x^J -* eo instead of |x| -* oo } together with the requirement that G have the property that G fl {x | |x n| <_.N}'. be bounded for any re a l N (Clark c a l l s such sets tubes along the x -axi s [8 ] ) . 2. The following conditions, due to Persson [ 2 8 ] , also imply that L i s bounded below: ( i ) i f 0 < C < F and £. > 0 i s ar b i t r a r y , then W ^ ' B P I G + 0 M U I I B R G f o r a n y U e C o ^ G ^ ' where c i s a constant depending on € , C and r ; and ( i i ) V Q = l i m (ess i n f V(x)) > -co . r-»co xeG - B r The l o c a l condition appears to be unrelated to A ( i ) , however the second condition, concerning behavior near co , implies A.(ii). We are now ready to f i n d the P.E. of L . For this we require: D e f i n i t i o n For a p o t e n t i a l V s a t i s f y i n g Condition A, denote by W^-'2(G,V) the completion of c"(G) with respect to the-norm i i u i i 2 v = l u i i 2 +r v i u i 2 d x . - i - , v J_ , c «. _ 1 3 . L e t t i n g B[u,v] = (vu,vv) + ((V + l)u,v) , we have H o (the completion of C"(G) with respect to B ) i s W^ 2(G,V) and B x[u,v] = (vu,vv) + ((V + l)u,v) / where derivatives are i n the weak sense. Also the F.E. i s L 1 = -A + V + I (weak sense). The domain of L^, £(L-^) , turns out. to be i> s W^ 2(G,V) 0 {u £ L_(G) | -Au + Vu e L 2(G)} ; 1 P for i f u e ->(L^) , then , u e W * (G,V) by construction of L-^  > and also there exists g e L^(G) with . L ^ g = u . But then, for any v e C~(G) , (g,v) = B - j L - ^ v ] = B 1[u,v] = = (vu,vv) + ((V + l)u,v) = (-Au + Vu + u,y) where the l a s t equality i s from the d e f i n i t i o n of A i n the weak sense. Therefore -Au + Vu + u = g e Lg(G) , and hence u e i> . Conversely, i f u e i> , then f o r any v e C™(G) , by the construction of L ^ we" have B 1[L~ 1(-Au + Vu + u),v] = (-Au + Vu + u,v) = B 1[u,v] Since B-^  i s an inner product on W^^2(G,V) , u = L-L"^(-Au + Vu + u)'. by completeness, so that u e i>(L-^) 14. We now c o n s i d e r v a r i o u s a d d i t i o n a l c o n d i t i o n s on V and G f o r w h i c h the above s e l f - a d j o i n t o p e r a t o r i s semi-d i s c r e t e . The q u e s t i o n o f n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s remains open. (a) Suppose V ( x ) +» as |x| -• » , x e G ; assume c o n d i t i o n A i s s a t i s f i e d . L e t L denote the F.E. o f -A + V + I on c >> • Theorem 1.2 I f the p o t e n t i a l s a t i s f i e s V ( x ) -• +». as |x| •* » j x s G j t h e n the spectrum of L c o n s i s t s o f an i n f i n i t e sequence o f i s o l a t e d e i g e n v a l u e s , > w i t h f i n i t e m u l t i p l i c i t i e s , h a v i n g \ -• » . The p r o o f i s g i v e n v i a t h r e e lemmas. Lemma 1.5 L e t S c L P ( G ) , p >_ 1 , where G c E n . Then S i s precompact ( i . e . S i s compact) i n L P ( G ) p r o v i d e d (a) f o r any € > 0 t h e r e e x i s t s an R such t h a t J | f | p d x < £ f o r G-B R any f e S ; (b) { f | ? | f e S} i s precompact i n L P ( F ) f o r any bounded s e t F i n G . P r o o f : S i n c e the p r o o f i s w ell-known, ( c f . [ 1 0 ] , [ l ] ) , we o n l y g i v e a s k e t c h . L e t ,{f } = {f } be a sequence from S and I I Tl y O G = U F± where F± c F ± + 1 and F± a r e bounded. D i a g o n a l i z e the sequences [ f .} where {f ' .} i s a subsequence o f n, j . n, j. {*V, J c o n v e r g i n g i n L P ( F . ) , j = 1,2,... . 15. Lemma 1.4 If V s a t i s f i e s the hypotheses of Theorem 1.2, then W^ 2(G,V) -+ Lg(G) i s a compact embedding. 1 2 Proof: Let {u^} be a bounded sequence i n WQ> (G,V) . We 2 v e r i f y (a) and (b) of Lemma 1.3*. l e t € > 0 and || u^ || 1 y _£ K . Since V(x) - <=° , there exists R > 0 with V(x) >_ 0 for |xj >_ R . By Condition A ( i ) , there exists a constant c(R) with J v j ^ ^ d x _> c J* l ^ ^ d x f o r any i . G n B R GT1B R Therefore K >_ ( J + J )V|u ±| 2dx >_ J V j u ^ d x + c J |u ± 12dxc , G^B^ GOB- G-B GHBT, R R r R for any r >_ R , and any i . But then J V|u. | dx _< "I ( K B K ( l + |c|) . Taking r large enough so that V(x) > K(l+|c for [x| >_ r > R , we have J |u. | dx <_ e f o r any i . G-B r To show (b), l e t P be a bounded set i n G so that p c Br, for some R . • Extend u. to be zero outside . G . Then ( U j j g ^ i s a bounded sequence i n W 1 , 2 ( B R ) , so that R W l j 2 ( B R n G) - L 2 ( B R n G) i s compact i f n > 2 , by a well-known theorem of Sobolev, R e l l i c h , Kondrachov (cf. [5]). Hence •(b) i s s a t i s f i e d . Therefore the embedding 1,2 ¥ Q ' (G,V) -• L^(G) i s compact. 16. Comment. A t . t h i s p o i n t t h e r e a r e a t l e a s t two a l t e r n a t i v e . p a t h s we c o u l d t a k e t o p r o v e t h e theorem. The s t a n d a r d method u s e s t h e L a x - M i l g r a m t h e o r e m and Lemma 1.4 t o show t h a t t h e r e s o l v e n t o p e r a t o r , ' ( L - ( i l . ) - 1 , i s c o m p a c t . f o r any i-i i n t h e r e s o l v e n t s e t ( £ '(f)) .. The R I e s z - S c h a u d e r t h e o r y f o r compact o p e r a t o r s t h e n s h ows'the s p e c t r u m o f ( L - u l ) - 1 t o be an i n f i n i t e s e q u e n c e o f e i g e n v a l u e s , {u n} , t e n d i n g t o z e r o ; t h i s i n t u r n g i v e s t h e r e q u i r e d e i g e n v a l u e s , f ^ n ^ > ^ o r ^ (\ = H + — ) • F o r d e t a i l s o f t h i s method, s e e [ 5 ] , [ 1 4 ] , : r. [9 ] , e t c . . S i n c e we w i l l be i n t e r e s t e d i n v a r i a t i o n a l p r i n c i p l e s we p r o v e a weaker s t a t e m e n t w h i c h e x h i b i t s t h e e i g e n v a l u e s i n t h a t f o r m . We u s e t h e n o t a t i o n i n f f ( u ) [u e II] t o mean t h a t t h e i n f i m u m i s o v e r a l l u e U . Lemma -1.5 L e t B [ u , v ] = ( v u , v v ) + (Vu,v) + (u,v) f o r u,v e W^ j 2(G,V). and S = {u e W l j 2 ( G , V ) | || u || Q = 1} . ' Then = i n f B [ u , u ] [u e S] i s assumed, s a y a t ; i n g e n e r a l X n = i n f B [ u , u ] [u e S fl U^"-j_ ] i s assumed, s a y a t u n (n - 2 , 3 , . . . ) ; h e r e . = {u e L - ( G ) | ( u , u . ) Q = 0 ( j = l , . . . , n - l ) } . M o r e o v e r , t h e a r e t h e e i g e n v a l u e s o f L . P r o o f : L e t {f^} be a s e q u e n c e o f v e c t o r s i n W 1 ; , 2(G,v~) w i t h jf f n || Q = 1 and B [ f n , f n ] - \ 1 • S i n c e { f n } i s bounded i n W ^ 2 ( G , V ) and ..W^ 2(G,V) -> L p ( G ) i s compact (Lemma 1 . 4 ) , t h e r e e x i s t s a s u b s e q u e n c e , a g a i n d e n o t e d l?-^ > a n e l e m e n t 6 L _ ( G ) , w i t h f n -» . i n _ 2 ( G ) . I n f a c t 1 7 . u-, e ¥ ' (G.V) : l e t € be r e a l , k > 0 a c o n s t a n t and [ g n ^ b e a s e q u e n c e i n L 2 ( G ) h a v i n g || g n |].^ y <_ k .' Then B [ f n + 6 g n , f n + € g n ] > X^' If f n + € g n || 2 , so t h a t . ( B [ f n ? f n ] - X x ) + 6 2 ( B [ g n J . g n ] - X 1|| g n || 2) + € 2 R ( B [ f n , g n ] -X l ^ f n ' ^ n ^ o ^ - 0 * F o r s i m P l i c i t y ^ A n + ^ B n + € C n >_ 0 , where 0 <_ A n -> 0 as n - °° , and 0 <_ B n <_ K . L e t t i n g € = + JA ( i f A = 0 t h e n C = 0 and we a r e done) we see — v n v n n ' t h a t |C | <_ M JA^ where M depends o n l y on K . U s i n g ( i g n 3 t h i s shows t h a t f o r any 5 > 0 , t h e r e e x i s t s N(6,K) s u c h t h a t n > N i m p l i e s Nf^ ,^ ] - x 1 ( f n . g n ) 0 l < 5 f o r a n Y i s ^ h a v i n g . || g n - l ^ y <. K . . T h e r e f o r e , u s i n g f i r s t g n = f n - f m and t h e n g_ = f - f , we o b t a i n n m | B [ f - f , f - f ] - X. || f - f I'|2|. - 0 as n,m - » . F i n a l l y , I n m 5 n m 1 " n m " o 1 . ' I! f - f |L2 T r = B [ f - f , f - f ' ] < [ B [ f - f , f - f ] -II .n m "1,V n m} n m — 1 n m' n m X l " f n ~ f m " o 1 + X l . " f n _ f m " o "* 0 s i n c e { f n } ' i s C a u c h y i n L g ( G ) . B y . c o m p l e t e n e s s and u n i q u e n e s s o f l i m i t s , f n - u l 6 W o ' 2 ( G * V ) • S i n c e B [ f n , f n ] - B f u ^ u ^ = || f R | | 2 ^ - [| u± we have B[u-^,u^] = X . A p p l y t h e same p r o c e d u r e i n t h e s u b s p a c e s o r t h o g o n a l t o t h e d i s c o v e r e d e i g e n f u n c t i o n s t o g e t t h e n e x t u ^ .• By Theorem 2.3 and Lemma 2.8, X^ i s an e i g e n v a l u e o f L . 18. (b) Suppose V ( x ) i s bounded. T h i s case ( w h i c h i n c l u d e s the L a p l a c i a n ) c l e a r l y has - A + V + I bounded below on C™(G) > and the t h e o r y f o r the F.E.., L , t o be s e m i - d i s c r e t e i s the same as i n (a) i f we can o b t a i n compactness o f t h e embedding W^ j 2(G) - L 2 ( G ) . For t h i s i t i s n e c e s s a r y t h a t G " s h r i n k a t co " i n the sense t h a t d i s t ( x , 3 G ) -• 0 as |x| -> co . x e G ( s u c h s e t s w i l l be c a l l e d , a c c o r d i n g t o s t a n d a r d usage, q u a s i -bounded ( c f . d e f i n i t i o n p. 2 3 ) ) : f o r i f G c o n t a i n e d an i n f i n i t e sequence of congruent b a l l s o f f i x e d r a d i i , t hen the t r a n s l a t e s , [ f n 3 , o f a C°°-function w i t h s u p p o r t i n one o f the b a l l s would have |[ f |L 0 = c and || f - f || .••= c 1 , thus i l _L j i l Til. O JL p r e v e n t i n g compactness. S e v e r a l c o n d i t i o n s on G a r e known f o r compactness o f the embeddings; one o f the most r e c e n t i s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n due t o Adams [ 3 ] . . F o r t h i s , l e t H be an n - d i m e n s i o n a l cube o f s i d e h and E a c l o s e d s u b s e t . o f H . Denote by CC°(H,E) the c l a s s o f C r o(H) f u n c t i o n s w h i c h 1 2 v a n i s h n e a r E . The W ' - c a p a c i t y o f E i n H I s g i v e n by I. h 2 H J |D au| 2dx C ^ 2 ( H , E ) - i n f : | a U \ , , p H : [u e C-(H,E)] . P u p d x 1 2 Then the embedding ¥ q ' (G) - L (G) i s compact i f and o n l y i f f o r e v e r y 6 > 0 t h e r e e x i s t s h <_ 1 • and r 0 such t h a t e v e r y n-cube. H• o f s i d e -..h-, h a v i n g . H D {x e G | • [x| >_ r} jL $ } s a t i s f i e s C ^ 2 ( H , H-G) _ h ^ / 6 . 1 9 . I n o r d e r t o s h e d g e o m e t r i c l i g h t on t h e n o t i o n o f c a p a c i t y , and t o show how t h e d i m e n s i o n o f t h e b o u n d a r y i s i n v o l v e d , Adams a l s o p r o v e s , [ 4 ] , t h a t t h e r e e x i s t s a c o n s t a n t K s u c h t h a t f o r e v e r y c u b e H h a v i n g s i d e h <_ 1 , C ^ 2 ( H , H-G) >_ K (H,G) h w h e re u n _ ^ ( H , G ) i s t h e maximum, t a k e n o v e r a l l p r o j e c t i o n s P o n t o ( n - 1 ) - d i m e n s i o n a l f a c e s o f H , o f t h e ( n - 1 ) - m e a s u r e o f P(H-G) . A n a l o g o u s l y t o p a r t ( a ) we h a v e Theorem 1.6 I f V ( x ) i s b o u n d e d and G s a t i s f i e s t h e a b o v e c o n d i t i o n , t h e n t h e s p e c t r u m o f L i s d i s c r e t e . I f G does n o t s h r i n k a t co 3 t h e n i t i s s t i l l p o s s i b l e t o g i v e some b o u n d s f o r t h e , d i s c r e t e p a r t o f t h e s p e c t r u m o f L ( o u r r e s u l t s h e r e w i l l no l o n g e r r e q u i r e V t o b e b o u n d e d , , B °: b u t o n l y r e q u i r e V ( x ) -» V q a s |x| -*» and C o n d i t i o n A t o b e s a t i s f i e d ) . F o r t h i s , l e t d L . d e n o t e t h e l a r g e s t r e a l number, p o s s i b l y co 3 b e l o w w h i c h t h e s p e c t r u m o f L i s d i s c r e t e o r empty. Lemma 1.7 I f V s a t i s f i e s C o n d i t i o n A, so t h a t -A + V + I i s b o u n d e d below''on C Q ( g ) > t h e n d L = t = l i m i n f ( L f , f ) r-»co [ f e ' C * ( G - B r ) , H f J) = 1 ] , w h e r e L i s t h e P.E. o f -A + V + I . 2 0 . P r o o f : F o r any s e l f - a d j o i n t , o p e r a t o r L , the p r o o f t h a t . d L — S i v e n by P e r s s o n [28]; t h e r e f o r e we need o n l y show _ . I f d_^  = co , we a r e done, so assume d^ < => , L e t C(L) denote the n o n - d i s c r e t e p a r t of the spectrum o f L By d e f i n i t i o n , d L = i n f {X | X e C(L)} .• We show ^ < \ f o r any X e C(L) . L e t X e C(L) and £un3 e -^(L) be a c h a r a c t e r i s t i c sequence; ( i . e . || L u n - X u n ||_ - 0 , II u n l l Q = 1 a n d u -» 0 weakly ( c f . [36])). B e f o r e p r o v i n g lr < X , we n j_/ r e q u i r e some f a c t s about { u n 3 • F i r s t , f o r any R > 0 , t h e r e i s a c o n s t a n t K such t h a t S (V- + l ) | u n | 2 d x BRflG - II' L% llo,BpflG l'l;un " c B^G " C( R) J* l ^ l ^ x 1 ^ + |C; R R BRno by C o n d i t i o n A ( i ) .and the p r o p e r t i e s o f t u n 3 • Next we show t h a t " !l U n llo,B_,nG " 0 a n d (I 7' V U n V n G " 0 ' a S n " * * : R • R 2 W^ J , 2(GnB R) - L 2 ( G T 1 B R ) i s compact ( R e l l i c h ) so f o r some sub-sequence, {u^} , || u ^ [|2 B n G •-» 0 ( u ^ converges and u^- 0 ' R w e a k l y ) . But then £u n) . has t h i s p r o p e r t y ( o t h e r w i s e we get 2" II u n II1 B nG — K + 1 ^ t h e a t , o v e r e s u l t , b u t 21. a c o n t r a d i c t i o n u s i n g t h e same argument on a s u b s e q u e n c e ) . L e t t i n g V + = {x e G | V ( x ) _> 0} , V~ = ( x e G | V ( x ) £ 0} , we have 0 < J |V| f u n | 2 d x = J V | u n | 2 d x + ./ - V | u n | 2 d x ' B R O G B R n v+ B R n v ~ By C o n d i t i o n A ( i ) , t h e s e c o n d t e r m i s bounded b y | C(S) | || u IJ . ,, N P , hence i t a p p r o a c h e s 0 b y t h e above. ll O ^ X J ' »-T F o r t h e f i r s t t e rm, S V | u n [ 2 d x < ( L u n , u n ) B n y + < 'I L u n "o,BRnGH u n ^ o , B R H G B R n v + • R < ( I M I K H o , B R n G + ^ D ) l [ % f l 0 ^ B R n G , w h i c h a g a i n a p p r o a c h e s 0 . T h e r e f o r e ( l v l u n J u n ) g f)G ~* 0 R N e x t l e t 0 < r < R , and l e t f e C c ° ( E n ) be d e f i n e d b y x e B r , and h a v i n g | v f ( x ) | bounded i n G . Then f u i s i d e n t i c a l l y 0 I n B R and f u n e ¥ ^ 2 ( G , V ) ( i f V - U r . as m - co i n w J ' 2 ( G 3 V ) w i t h Vn>m e C-(G) , t h e n f V m - f u n i n 2 2 . W ^ 2 ( G , V ) ) . N o t i c e t h a t I! f u ||2 - 1 as n -» » " n 11 o ( | [ f u n | | 2 - J | f u j 2 d x + / | u n | 2 d x - . 0 + l s i n c e G r V G - B R N ^ ' l o , B R n G ^ 0 a n d I K I I " = D and || f u n | | 2 ^ < X|| u n ||2 + o ( l ) (|| f u n | | 2 j V = || V ( f u n ) ||2 + || f u n ||2 + (V f u ' , f u n ) < (I v u n I I 2 . + o ( l ) + H u n [ j ' 2 + ( V u n , u n ) + o ( l ) 2 2 and II u n H ^ y "* M l % II 0 b y t h e p r o p e r t i e s o f u n ). F i n a l l y we p r o c e e d t o show f o r € > 0 t h a t lL - € < X : t h e r e e x i s t s r > 0 w i t h ( L f , f ) > || f ||2 f o r a n y f eC^G-B^,) , b y t h e d e f i n i t i o n , o f lT . By o r ±j 1 2 c o n t i n u i t y t h i s i s t r u e f o r a n y f e W J (G,V) , supp f e G - B r . L e t R > r . Cho o s e f as a b o v e , so t h a t ( * L ~ € ) , , f U n , ! o < " f U n " l , v l M | u n | | 2 + o ( l ) =X + o ( l ) . S i n c e || f u ||2 - 1 , lT - e < X . 11 n 11 o L — I n o r d e r t o g i v e b o u n d s f o r t h e d i s c r e t e p a r t o f t h e s p e c t r u m o f L , we u s e t h e f o l l o w i n g c l a s s i f i c a t i o n o f u n b o u n d e d s e t s due t o G l a z m a n [ 1 5 ] . D e f i n i t i o n A s e t G c F/1 w i l l be c a l l e d q u a s i - c o n i c a l i f a n d o n l y i f i t c o n t a i n s a r b i t r a r i l y l a r g e b a l l s . I f G i s n o t q u a s i - c o n i c a l , b u t c o n t a i n s a n ' i n f i n i t e number o f d i s j o i n t b a l l s w i t h f i x e d p o s i t i v e r a d i u s , t h e n i t w i l l b e c a l l e d q u a s i - c y l i d r i c a l . I n s u c h a c a s e , t h e r a d i u s o f G 23. i s d e f i n e d as the supremum of a l l such r a d i i . I f G i s unbounded, b u t n e i t h e r q u a s i - c o n i c a l n o r q u a s i - c y l i n d r i c a l , then G w i l l be c a l l e d quasi-bounded. C l e a r l y G I s quasi-bounded i f and o n l y i f d i s t (x,3G) -* 0 as |x| -» o= } x e G . Theorem 1.8 L e t G be q u a s i - c o n i c a l . I f V ( x ) s a t i s f i e s C o n d i t i o n A (V c o u l d be bounded) and V ( x ) -» V"0 as [x| -» co , x e F , f o r a q u a s i - c o n i c a l s u b s e t F o f G , t h e n ^ = V . P r o o f : The p r o o f f o l l o w s from Lemma 1.7 as i n [28] ( P e r s s o n ) . For q u a s i - c y l i n d r i c a l domains we r e q u i r e the f o l l o w i n g c o n d i t i o n , due t o Adams [2]: t h e r e e x i s t s R Q _ 0 such t h a t f o r e v e r y R >_ R Q t h e r e c o r r e s p o n d p o s i t i v e numbers d('R) and 6(R) h a v i n g ' d n + 1 ( R ) 6 1 - n ( R ) <_ M f o r a l l R , and s a t i s f y i n g the p r o p e r t y t h a t f o r each x e G - B R , the b a l l B-^(X) i s d i s c o n n e c t e d I n t o two open components C-^  and C^ by a subset S ^ of 3G i n such a way t h a t C fl: B ^ ( x ) c o n t a i n s a b a l l o f r a d i u s 6(R) f o r i = 1,2 . Theorem 1 . 9 L e t G be q u a s i - c y l i n d r i c a l w i t h r a d i u s a . I f V s a t i s f i e s C o n d i t i o n A and V('x) -» V Q as j x j °= , then V 0 <_ d L _ V o + \ ( a ) , where X ( a ) i s the s m a l l e s t D i r i c h l e t e i g e n v a l u e o f -A on B (©) . I f i n a d d i t i o n G s a t i s f i e s the above c o n d i t i o n then the l o w e r bound may be 24. improved t o d^ _ V Q + ( c M ) " 1 , c b e i n g the c o n s t a n t i n ( l ) i n the p r o o f . P r o o f : The f i r s t statement f o l l o w s from Lemma. 1 . 7 as i n [ 2 8 ] ( P e r s s o n ) . Suppose the hypotheses o f the second statement h o l d . Then ( c f . [ 2 ] ) , the f o l l o w i n g "Poincare''-type" i n e q u a l i t y h o l d s : there e x i s t s a c o n s t a n t c } depending on n , such t h a t f o r a l l r > R , and a l l f u n c t i o n s u e ^o' '1) f |u|^dx < c d ( r ) n + 1 6 1 _ n ( r ) f |vu|^dx . G-B r . • G Hence i f R Q i s l a r g e enough so t h a t J v ( x ) - V Q | < € , r_>_R Q , then f o r any u e C ~ ( G - B r ) , [| u || = 1 , we have (Lu,u) _ II vu ||2 + (Vu,u) >_ ( c M ) " 1 + V Q - € , so t h a t lT = i n f ( L u , u ) [ u e . C~(G-B r) , || u ||_ = l ] _> ( c M ) - 1 + Vq - € . F i n a l l y d L = l i m iT _ ( c M ) " 1 + V Q . r-»co (c) (V(x) -> -co) Suppose V ( x ) -» -oo as |x[ -* co } x e G . I f t h i s o c c u r s a t a r a t e comparable t o C(x) , t h e n i t i s s t i l l p o s s i b l e to have -A + V + I bounded below u s i n g C o n d i t i o n A . P e r s s o n ' s c o n d i t i o n s do n o t p r o v i d e f o r . such b e h a v i o u r of V . I f C( x) ~* 0 as |x| -> oo } then G i s q u a s i -bounded; the c o n v e r s e i s f a l s e ( f o r example pepper s e t s c o u l d e a s i l y be c o n s t r u c t e d w h i c h do n o t have C ( x ) -* 0 as |x| - co). Theorem 1 . 10 I f -V • s a t i s f i e s C o n d i t i o n A , and I n a d d i t i o n — 2 i n f [-^|^— + V ( x ) ] [x £ G n B R ] ^ co as |x| = R - then the spectrum of the F.E.', L , o f -A + V + I on CQ( G) I S d i s c r e t e . P r o o f : -A + V + I i s bounded below so t h a t L e x i s t s . — 2 F o r g i v e n N f i n d r such t h a t — + V ( x ) >_ N , |x| >_ r . Then f o r any f e C*(G-B ) , || f |J = 1 , we have ' o r ' 2 ( L f , f ) >_ J" [ f r (-s— + V)dx by Lemma 1 .1 , so t h a t ( L f , f ) >_ N G C Hence d-^  >_ -L^ >_ N . by Lemma 1 .7-Remarks: 1-. P e r s s o n [ 2 8 ] , and C l a r k [8] g i v e r e s u l t s con-c e r n i n g d i s c r e t e n e s s o f L on quasi-bounded domains, however they r e q u i r e f u r t h e r r e g u l a r i t y c o n d i t i o n s f o r t h e boundary. 2. The above t h e o r y g e n e r a l i z e s some o f the known r e s u l t s c o n c e r n i n g the spectrum o f the S c h r o d i n g e r o p e r a t o r on t h e whole space, E^ , and on bounded s e t s , G , by s p e c i a l con-s i d e r a t i o n s of C(x) ( c f . [8] and the b i b l i o g r a p h y t h e r e ) . ( i v ) G e n e r a l Second Order O p e r a t o r s . I f L i s a symmetric second o r d e r o p e r a t o r , t h e n 1 2 the F.E. would have domain i n s i d e W ' (G) i f the c o e f f i c i e n t s o v were u n i f o r m l y bounded; much more g e n e r a l , c o n d i t o n s on t h e c o e f f i c i e n t s c o u l d be g i v e n f o r L t o s a t i s f y G a r d i n g ' s i n e q u a l i t y ( c f . [ 5 ] , [ 1 1 ] ) so t h a t some o f the above t h e o r y would c a r r y o v e r . CHAPTER TWO VARIATIONAL PROPERTIES A. V a r i a t i o n a l P r o p e r t i e s f o r S e m i - d i s c r e t e O p e r a t o r s I n t h i s s e c t i o n we a r e concerned w i t h the e i g e n -v a l u e s o f s e m i - d i s c r e t e o p e r a t o r s and t h e i r v a r i a t i o n a l ' p r o p e r t i e s . I f L i s a compact s e l f - a d j o i n t o p e r a t o r , w i t h domain ^ ( L ) C : H , and X^ <_ Xg <_ ... a r e the e i g e n v a l u e s o f L , w i t h 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 s u-^,Ug,..., t h e n i t i s w e l l known t h a t they s a t i s f y the R a y l e i g h c h a r a c t e r i z a t i o n : X n = min ( L u , u ) [ u e S fl U ^ _ 1 J (n = 1 , 2 , . . . ) where S = [u e £(L) | || u || = 1}. and U1. = [u e H | (u,u. ) = 0 , 1 = l , . . . , j } " . A l s o w e l l known i s t h e f a c t t h a t the e i g e n -v e c t o r s a r e n o t r e q u i r e d f o r a v a r i a t i o n a l f o r m u l a t i o n : X n = max min (Lu,u) [P e P n _ 1 | u e S fl p 1 ] , (n = 1 , 2 , . . . ) , where the n o t a t i o n means t h a t the maximum i s t a k e n o v e r a l l n - 1 d i m e n s i o n a l subspaces P cz Lg(G) , and t h e minimum i s o v e r a l l u e S D {u e H | (u,p) = 0 f o r any p e P] , S as above. These f a c t s can be e a s i l y found ( c f . [±J>]). I f L i s an unbounded s e m i - d i s c r e t e o p e r a t o r w i t h e i g e n v a l u e s X^ <_ Xg <_ . . . , then the above c h a r a c t e r i z a t i o n s remain v a l i d ( c f . [ 3 1 ] ) - S e v e r a l c o n d i t i o n s and t e c h n i q u e s a r e known wh i c h t e l l when s e l f - a d j o i n t o p e r a t o r s have e i g e n -v a l u e s - see Chapter One f o r example. We d i s c u s s a n o t h e r c o n d i t i o n , namely the c o n v e r s e to t h e above v a r i a t i o n a l s t a t e -ments:, when a r e the numbers X-, = i n f (Lu,u) [u e S] , 27. X n = sup i n f (Lu,u) [ P- e P n ^ J u e S n P"*"] (n = 2,3,.-.) i n f a c t e i g e n v a l u e s o f L ? I f the X 's a r e assumed, then i t i s known t h a t . t h e y a r e e i g e n v a l u e s ( c f . M i k l i n [25]). We g i v e a c o n d i t i o n , analogous t o t h a t o f B e r k o w i t z , [6], a s s u r i n g t h i s . An easy s e t t h e o r e t i c argument (Lemma 2.1) shows t h a t { X n l i s a n o n - d e c r e a s i n g sequence, so t h a t we.may c o n s i d e r X = l i m X (w h i c h may be i n f i n i t e ) . U s i n g the s p e c t r a l n-»co t h e o r y f o r L , i t i s not h a r d t o show t h a t X^ s e p a r a t e s the d i s c r e t e and c o n t i n u o u s spectrum o f L (Theorem 2.2). I n f a c t , i f X n < X r o , then X-^ .,...,X a r e t h e f i r s t n e i g e n v a l u e s of L ( u s i n g the s t a n d a r d p r o c e d u r e ' o f r e p e a t i n g an e i g e n v a l u e p times i f t h e di m e n s i o n of i t s e i g e n s p a c e i s p ). Lemma 2.1 L e t L be s e l f - a d j o i n t w i t h domain i> a H and se t X-j_ = i n f (Lu,u) [u e S] , X n = sup i n f (Lu,u) [ P e P ^ ^ u e S fl P^"-] (n - 2,3,...) where S , P"1, Pr_-]_ a r e a s above. Then X n < X n + 1 (n - 1,2, . . . ) .. P r o o f : C l e a r l y X 1 <_ X g . For n >_ 2 , l e t P e P n _ 1 Q. = P © [v] f o r some v | P . Then Q e P n and S n Or cz s n P 1 , so t h a t X n + 1 >_ i n f (Lu,u) [u e S n Q 1] _ i n f (Lu,u) [u £ S fl P v ] f o r any P e P n _ 1 . T h e r e f o r e Xn+1 > Xn • and Theorem 2.2 [6]. F o r the numbers C^n3 d e f i n e d i n Lemma 2.1, l e t X = l i m X„ . co n n-»oo 28. (a) I f X^ < X , t h e n X X _ a r e the f i r s t n e i g e n -\ / n «> . I n v a l u e s o f L ; (b) I f u < X^ , then the spectrum o f L i s d i s c r e t e below u (c) I f i_i > X^ , then the spectrum o f L i s n o t d i s c r e t e below u . P r o o f : A l l the p r o o f s w i l l use the r e s o l u t i o n o f i d e n t i t y , ( E ( u ) } a s s o c i a t e d w i t h L . To show '(b), f i n d n such t h a t U < X n + 1 • We w i l l show dim (E(u)H) <_ n , so t h a t the spec-trum o f L below u c o n s i s t s of a t most n e i g e n v a l u e s . The p r o o f i s by c o n t r a d i c t i o n . Suppose dim (E(u)H) > n . L e t P be any subspace of H w i t h d i m P = n . Then t h e r e e x i s t s u e E.(u)H fl p \ w i t h (Lu,u) _< u|| u ||2 by t h e s p e c t r a l theorem. S i n c e E ( u ) H <= } we have i n f ( L u , u ) [ u e S fl P 1 ] <_ f o r any such P . T h e r e f o r e ^n+j_ _S ^ > a c o n t r a d i c t i o n . To show ( a ) , ' o t n o t e t h a t u < X n i m p l i e s dim (E(u)H) <_. n - 1 by the above p r o o f . S i n c e X n < X^ , (b) a l s o shows t h a t dim (E(X )H) < » . T h e r e f o r e i n o r d e r t o t h c o n c l u d e t h a t X i s an e i g e n v a l u e ( c l e a r l y t h e n ), i t s u f f i c e s to show dim (E(u)H) >_ n f o r any u > X n . F o r t h i s we a g a i n use a c o n t r a d i c t i o n argument. Suppose dim (E(u)H) = k < n . Now f o r u e ( E ( u ) H ) 1 , we have (Lu,u) >_ u|| U |f by the s p e c t r a l theorem, so t h a t i n f (Lu,u) [u € S n ( E ( u ) H ; ] >_ u > X n . But then X k + 1 = sup i n f (Lu,u) [P e P^ | u e S n P" 1] >_ i n f (Lu,u) [u e S fl ( E ( u ) H j " ] > X , a c o n t r a d i c t i o n . 2 9 . By the above p r o o f s , dim [E(X r o+€)H - E(X (_-e)H]= » f o r any 6 > 0 . T h e r e f o r e (c) i s p r o v e n ( c f . [ 1 9 ] ) -B. V a r i a t i o n a l P r o p e r t i e s f o r B i l i n e a r Forms. I n our d i s c u s s i o n 'of t h e examples of Chapter One, r e c a l l t h a t w i t h each symmetric semi-bounded o p e r a t o r L t h e r e was a s e l f - a d j o i n t e x t e n s i o n (F.E. ), , d e f i n e d on .#(1^) , and a serni-bounded b i l i n e a r f o rm B-^  d e f i n e d on the space H Q , where 2>(_•]_) £ H q . I n c o n n e c t i o n w i t h the e i g e n v a l u e s o f we have the f o l l o w i n g Theorem 2 . 5 L e t L• be a symmetric o p e r a t o r w i t h _?•(—) dense i n H Q . I f B i s a semi-bounded form w i t h B [ f , g ] = ( L f , g ) f o r any f , g e .#(_) and B-^  i s the e x t e n s i o n by c o n t i n u i t y o f B t o H , t h e n X i s an e i g e n v a l u e of L-^  , the F.E. o f L , w i t h e i g e n f u n c t i o n u i f and o n l y i f u e R"o and B 1 [ u , v ] = X(u,v) f o r any v e H Q . P r o o f : F i r s t l e t u e -^(L^) , X e $ such t h a t L-^u = Xu . Then u e H Q , so t h a t (Xu,v) = (L^u,v) = B-^[u,v] f o r any v e H D C o n v e r s e l y , suppose u e H q and B^[u,v] = X(u,v) f o r any v e H Q. . Then B-^tUjV] = B-^  [ L ' - ^ X u ) , v ] f o r any v e H Q . S i n c e H q i s complete w i t h r e s p e c t t o B^ , we have L ^ 1 ( X u ) =.u or u e i>(L^) and L-^u = Xu . 3 0 . I n v i e w o f t h i s , r e s u l t we make the- f o l l o w i n g : D e f i n i t i o n s 1. I f t h e b i l i n e a r form B i s d e f i n e d o v e r some H i l b e r t space ¥ , and t h e r e e x i s t X e (j? , u e ¥ such t h a t || u [| = 1 and B[u,v] = X(u,v) f o r a l l v e ¥ , the n we c a l l X an e i g e n v a l u e o f B w i t h 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 u . 2 . I f ¥ . i s a subspace o f a H i l b e r t space H , and B i s d e f i n e d on ¥ , then B i s s a i d t o be c o e r c i v e over ¥ i n H i f t h e r e e x i s t s a c o n s t a n t c > 0 such t h a t B[u,u] >_ c|| u || f o r any u e W ( c f . [ 5 ] ) . Remark Theorem 2 . 3 c o u l d be r e s t a t e d as f o l l o w s : f o r any c o e r c i v e form B , t h e r e i s a s e l f - a d j o i n t o p e r a t o r h a v i n g the same e i g e n v a l u e s as B o r some e x t e n s i o n (by c o n t i n u i t y ) o f B . The same q u e s t i o n as i n S e c t i o n A a r i s e s : suppose H Q and B [ - , - ] a r e d e f i n e d so t h a t B i s c o e r c i v e o v e r Eq i n L p ( G ) , H Q i s complete w i t h r e s p e c t t o the i n n e r p r o d u c t d e t e r m i n e d by B , and so t h a t t h e r e i s a s e l f - a d j o i n t o p e r a t o r L h a v i n g the same e i g e n v a l u e s as B and h a v i n g -&(L) dense i n H Q ; when a r e the numbers | i 1 = i n f B[u,u] [U e S] } u n = sup i n f B [ u , u ] [ P £ p n _ i l ' u € S f l R 1 0 ] (n = 2,3,...)' 3 1 . e i g e n v a l u e s o f B , w h e r e now S = {u e H Q [ |[ u ||'Q = 1 } ( i n s t e a d o f [ u e _>(L) | || u ||q = l } ) a n d P n_ 1 i s t h e s e t o f n - 1 d i m e n s i o n a l s u b s p a c e s o f H q ( i n s t e a d o f L g ( G ) ) a n d = [ u € H Q | ( u , p ) Q = 0 f o r a n y p e P} ? S i n c e _>(_) i s d e n s e i n H Q i n t h e m e t r i c f r o m B [ - , - ] , = X-^ ; b y a lemma o f G o k h b e r g - K r e i n [16], 2>(L) n P^ 0 i s d e n s e i n H Q fl P ^ 0 . f o r a n y P e P n _ 1 , so |_n = sup i n f B [ u , u ] [ P e P n _ 1 | u e _>(L) 0 , || u |I = 1 } < \_a . Thus one s u s p e c t s ^n = ^n ' f > o r a 1 1 n " l f ^ n ^ s a s s u m e ( ^ "then t h i s i s t h e c a s e , b e c a u s e u n t u r n s o u t t o b e a n e i g e n v a l u e o f B (Lemma 2 . 8 ) , h e n c e o f L , so t h a t P n = X n b y S t e n g e r ' s r e s u l t , [ 3 1 ] • A n a l o g o u s t o S e c t i o n A, a c o n d i t i o n i s g i v e n ( T h e o r e m 2 . 4 ) t o i n d i c a t e when u n i s assumed. S i n c e t h e s p e c t r a l t h e o r e m i s n o t a v a i l a b l e , t h e p r o o f o f Theorem 2 . 4 i s b y f i r s t p r i n c i p l e s a n d c o n s e q u e n t l y l o n g e r . The o n l y t o o l s r e q u i r e d a r e i n d u c t i o n a n d some e l e m e n t a r y f a c t s o f g e o m e t r y i n H i l b e r t s p a c e s . The p r o o f i s . r e s t r i c t e d t o r e a l v a l u e d f u n c t i o n s p a c e s ( w i t h r e a l , s c a l a r s ) , h o w e v e r t h e p o i n t s p e c t r u m f o r s e l f - a d j o i n t o p e r a t o r s L , w i t h r e a l c o e f f i c i e n t s , i s t h e same w h e t h e r one u s e s r e a l o r c o m p l e x s p a c e s , so t h e , r e s u l t s a r e n o t a f f e c t e d f o r t h e o p e r a t o r s c o n s i d e r e d I n C h a p t e r One. L e t L g ( G ) , H Q b e r e a l s p a c e s w i t h B [ - , - ] c o e r c i v e o v e r H Q i n L ^ ( G ) . I t d e t e r m i n e s a n i n n e r p r o d u c t on H Q , d e n o t e d (•, • ) , w i t h c o r r e s p o n d i n g norm, || • || . 32. The p r o o f o f Lemma 2 . 1 shows U n <_ U-n+-|_ s o t h a t we may c o n s i d e r u = l i m u n n-»co Theorem 2.4 I f ^ < ^ 5 then u^, . . . , p n a r e the f i r s t n e i g e n v a l u e s o f B ( r e p e a t e d a c c o r d i n g t o m u l t i p l i c i t y ) . I n t h i s e v ent, the c l a s s i c a l R a y l e i g h c h a r a c t e r i z a t i o n i s v a l i d : = min B[u,u] [U e S fl U^"0^] ( i = l , . . . , n ) , where U. = [u..,...,u.] i s the e i g e n f u n c t i o n space. C o n v e r s e l y , i f = i n f B[u,u] [ U e S] i s assumed a t v^ , and a = i n f B[u,u] [u e S. fl _ ] i s assumed a t v (n = 2,3,...) n ; • n - 1 n . v > ' where V = [v.. , . . . ,v , ] , then a = U f o r a l l n . The c o n v e r s e statement i s a s t a n d a r d argument, so i s g i v e n as a s e p a r a t e lemma f o l l o w i n g t he r e s t o f the p r o o f (Lemma 2.10). We use i n d u c t i o n , so w i l l need t o keep t r a c k o f the e i g e n v e c t o r s amd m u l t i p l i c i t i e s . T h e r e f o r e we go to a more p r e c i s e (and awkward) theorem. F o r t h i s , we n o t e t h a t the statement u n < i s e q u i v a l e n t t o the statement t h a t the m u l t i p l i c i t y o f u n , say k^ , i s f i n i t e , i . e . ^ n + V 1 < ^ n + * n W h 6 r S ° n = m i n { i ' ^ i = ^ n } • U s i n S these f a c t s i t w i l l be more th a n enough t o p r o v e Theorem 2.5 F o r any n = 1,2,..., i f k n < c o J t h e n u-, ,. . . •, _ 1 a r e e i g e n v a l u e s o f B w i t h e i g e n v e c t o r s , Jn JI • {u-|,... su. , ' } c S , h a v i n g (u. ,u ) = 6 , and v e S n n~ "• ^ 33. has B [ v , v ] = u i f and o n l y i f v e [u . ,...,u , __] . • • J n J n \L A l s o ^ i = i n f B[u,u] [u e S 0 U ^ ] ( i = 2,...,_ n+k n) . The p r o o f i s g i v e n a f t e r f o u r lemmas. Lemma 2 .7 1 , u s i n g the g e o m e t r i c p r o p e r t y o f Lemma 2 . 6 , g i v e s the b a s i c e x i s t e n c e p r o o f f o r the f i r s t e i g e n v e c t o r : i f u 2 > , then the e x i s t e n c e o f a v e c t o r u-^  e S w i t h B[u-^,u^] = u^ i s shown. The same p r o o f i s a p p l i e d l a t e r i n o r t h o g o n a l sub-spaces t o f i n d the o t h e r u^'s . Lemma 2 . 8 shows t h a t the v e c t o r s u^ thus found a r e e i g e n v e c t o r s f o r X^ . Lemma 2 . 9 c o n t a i n s the i n d u c t i v e s t e p . Lemma 2 . 6 L e t x-^,x 2,x^ determine a C a r t e s i a n c o o r d i n a t e 3 system i n R . F o r x = S a.x. , s e t P x = a^x-, . Denote the u-sphere t h r o u g h the o r i g i n by S = {x | || x || = u} and f o r any b l e t T-^ denote the p l a n e p a r a l l e l t o the  X2. X2~ p l a n e , t h r o u g h [0,0,b] . F i n a l l y l e t a,a,6 be c o n s t a n t s / 2 2~ w i t h a > 0 , 0 < a < u , and 2,/\x -a >_ € > 0 . Then t h e r e e x i s t s 6 > 0 such t h a t the f o l l o w i n g h o l d s ( c f . F i g . l ) ' 3 i f E i s any e l l i p s o i d i n R^ c e n t e r e d a t 0 , w i t h d i s t (0,E fl TQ) > u + a , and w i t h two p o i n t s y ^ , y 2 e E h a v i n g [| y± || _< u + 6 , d i s t ( y ± , T & ) £ 6 and || y x - y 2 || >_ € , then t h e r e i s a p o i n t x e E w i t h || x |[ < u . P r o o f : L e t 6,o,a,,u be c o n s t a n t s s a t i s f y i n g the above r e q u i r e m e n t s . F i n d n l a r g e enough t h a t ^ < -|- and < -g- . We w i l l show t h a t the e l l i p s e 2 2 E ( 5 ) : X l + X 2 = 1 ( c f . P i g . 2) p a s s i n g , t h r o u g h the (u+o-) 2 \ ( 6 ) 2 FIGURE 2. p o i n t s Y ]_(| n,p) , Y 2 ( ^ P ) ••» w h e r e P 2.= (^+ 5) 2 - ( f n ) 2 (so t h a t || Y ± || = U+6 , || Y 1_-Y 2 'II = | n ) has X(6) < u I f 6 > 0 i s s u f f i c i e n t l y s m a l l ; we a l s o show t h a t f o r any e l l i p s o i d E , 35. s a t i s f y i n g t h e hypotheses o f the lemma w i t h t h i s 6 , the c r o s s -s e c t i o n t h r o u g h O^y-^y^ deter m i n e s an e l l i p s e w h i c h has a p o i n t x , || x || < u. , upon comparison w i t h E(6) . F i r s t t o show 6 > 0 e x i s t s so t h a t X < u : on the c i r c l e o f r a d i u s p. , c e n t r e 0 , t a k e a p o i n t Qi^,^) h a v i n g 0 < q-^  < , q 2 > 0 . There i s an e l l i p s e t h r o u g h the p o i n t s Q (u+a^O) w i t h .centre 0 and major a x i s a l o n g x-^ . . C l e a r l y i t s m i n or a x i s has l e n g t h X < u . The p o i n t P(p- L,p 2) on t h i s e l l i p s e w i t h p-^  = -jj; n , P 2 > 0 g i v e s the e s t i m a t e f o r 6 2 2 namely + P 2 l 6 n 2 ( u ^ j ) 2 '.X' 2 2 2 1 where we s e t p^ = (u + 6 ) - P-j_ I f t h i s has 6 > 6 or 5 > | , replace i t by min Of,§3 W i t h t h i s 6 , c o n s i d e r any e l l i p s o i d E s a t i s f y i n g the h y p otheses. The c r o s s - s e c t i o n of E t h r o u g h O ^ y ^ y ^ i n t e r s e c t s the p l a n e wQ i n a l i n e I ( b o t h p l a n e s c o n t a i n 0) . The e l l i p s e , , t r a c e d out by t h i s c r o s s - s e c t i o n ( c f . F i g . 3) has major a x i s w i t h l e n g t h a t l e a s t 2(u+a) s i n c e the w i d t h a l o n g £ i s a t l e a s t t h a t l a r g e ( d i s t (0,E fl TQ) _> u + a) ; n e x t we show t h a t one o f t h e p o i n t s I. FIGURE 3-3 6 . y ^ , y 2 ( r e c a l l u <_ || y^ |[ <_ u + 6) i s a t l e a s t ^ n u n i t s from the minor a x i s Zg , so t h a t by symmetry w i t h r e s p e c t t o t h i s a x i s , t h e r e i s a n o t h e r p o i n t z on , of the same h e i g h t from Z^ and d i s t a n c e from 0 . By comparison w i t h E(6) we see t h e r e would be a p o i n t x e E-^  h a v i n g [| x [| < |a . To show d i s t (y^,Z ) >_ f o r one of the y^'s we use a c o n t r a d i c t i o n argument. Suppose d i s t ( y ^ Z g ) = d i s t (yj_jPj_) < -|- f o r p^ • on the Zg a x i s ( i = 1,2) . We f i r s t n o t e t h a t || P x - P 2 II i f s i n c e € < || y1-y'2_H < || J1~V1 \\ + Il-P 1-P 2 II + 'I p 2 ~ y 2 " - f h + ' l p l " p 2 'I W e n o W c o n s i d e r t h e two p o s s i b i l i t i e s f o r y-^yg : ( i ) th e y b o t h l i e on one s i d e of ; ( i i ) t h e y l i e on o p p o s i t e s i d e s of Z^ . Case ( i ) o c c u r s ( c f . F i g , 3 ) when t i n t e r s e c t s Z-^  I n a s m a l l a n g l e , because the y^'s a r e f o r c e d t o l i e . c l o s e t o t , the l i n e of I n t e r s e c t i o n o f the p l a n e r and the c r o s s - s e c t i o n • -a t h r o u g h 0,y-, ,y ( r e c a l l d i s t (T ,y. ) < 6) ; Case ( i i ) ' o c c u r s when the a n g l e i s l a r g e r ( c f . F i g . 5)- F o r ( i ) , we . o b t a i n ( c f . F i g . 4, where t h e shaded a r e a r e p r e s e n t s the r e g i o n JI. U s i n g € < 2*/u'~ - , 6 < ^  and — < ^ - we have H < £ n 2 P l - p 2 |I < u + -| - 7 u 2 - • (u 2-"! 2) <. u + | ( 2 n - l ) 2n 4n< 2n Thus c a s e . ( i ) l e a d s t o a c o n t r a d i c t i o n . Suppose y ^ y ^ l i e o n o p p o s i t e s i d e s of as i n case ( i i ) ( c f . F i g . 5 ) . Then one o f the p o i n t s , say y^ , i s a t l e a s t as c l o s e t o & as to the Z 2 ~ a x i s . . S i n c e FIGURE 5 . where the l a s t d i s t (y^,l) > a - 6 ( r e c a l l d i s t ( y ^ 7 ^ ) <_ 5 > we have d i s t ( y - ^ ) >_ d i s t (y^l) > a - 6 > | > | n two i n e q u a l i t i e s a r e by the c h o i c e o f n and 6 . Thus case ( i i ) a l s o l e a d s t o a c o n t r a d i c t i o n . Lemma 2.7 I f > u-j_ , the n t h e r e e x i s t s e S such, t h a t II ^. II 2 H i B t u-^U-J ). P r o o f : • L e t {w.} be a m i n i m i z i n g sequence f o r w. k 1 k W, € S • k • 1 We w i l l show t h a t (w^ .} has a con-v e r g e n t subsequence. By the d e f i n i t i o n o f u 2 , t h e r e e x i s t s 2 I g e H Q w i t h y = i n f || u || [u e S n [ g ] • ] > 1 N o t i c e 38. that [g] ° = [h^] for some h^ e H q since we can f i n d e H-^  s a t i s f y i n g (u,g) Q = (u,h^) for a l l u e H q . Let h = J\i-^ || H - 1 ^ so that || h || = J\i^ and (1) Y = i n f || u ||2 [u € S n [h] 1] > u x • For u e EQ we can write u = ah + h"^" ; l e t P denote the projection operator Pu = ah , and Q i t s orthogonal complement, Qu = . For the sequence {w^ } , l e t F w k = a k h ' t h e n || Pw^l) 2 <_ ||wk || 2 - shows that |a^[ <_ || w^  || - 1 as k —* co 3 so that there exists a , . II h | | 0 <_ |a| <_ 1 , and a subsequence, again denoted {a^} , having a. -• a . k I f a = 0 , then there exists k large enough so that n x <_ || wk||2 <\i±+e , ,|| Pw^ ||2 = |a k| 2|| h ||2 < € ; therefore J l Q w k " 2 < ^ 1 + g ,. < ?1 + 6 < h - + g - ^ llQ.w kH 0 H v F w k H o d - l l Pw k|| o) 2 ( i - a / e ) 2 as € - 0 . Since Q w k e S fl [h] , ( l ) i s vi o l a t e d . " M o This contradiction shows |a| > 0 . I f [a| = 1 , we have || wk - w^. ||2 = ||Pwk-Pw ^  ||2 + || Qwk - Q'w || - 0 because ' P'wk - a h and ||'Qwk|p = || wk||2 - || Pwk||2 - - |j = 0 . Therefore {.wk} i s a Cauchy sequence- i n the space H , so that the existence of u 1 39. i s g u a r a n t e e d . I f 0 < |a| < 1 , we p r o c e e d w i t h a c o n t r a d i c t i o n argument. Suppose t h e r e e x i s t s € > 0 w i t h || w^ , - w || >_ € f o r a l l k , j . F i n d the number 6 > 0 from Lemma 2.6 u s i n g £ (made s m a l l e r i f n e c e s s a r y , t o f i t the r e q u i r e d r a n g e ) , H , a = Y - (a > 0 by ( 1 ) ) , a = |a| • Now p i c k two p o i n t s w ^ - ^ w - j such t h a t || Pw^-ah || < 6 , || Bw.-ah || < 6 , and || w, || <_ u + 6 , || w. || <_ u + 6 . We may c o n s i d e r {h,w .,w, }' as a l i n e a r l y i n d e p e n d e n t s e t by the J iv - - - - - . f o l l o w i n g argument: suppose h = a-|_w • + a 2 w ^ • F i n d w^ s a t i s f y i n g t he same c o n d i t i o n s as w.,w, and w i t h . . . . . . . J IV e <_ || - w k || < || w^ - w. || , say. I f w^ e [w^w^] , the n " w £ " W k = I' P W L " P w k I'2 + 'I Q W £ ~ Q w k I'2 W i t h I' P wl " P w k I'2 1 ( 2 6) 2 a n d I' Q w-L " Q w k " 2 -[ ( u + 6 ) 2 - ( a - 6 ) 2 ) " 5 - ( u 2 - ( a + 6 ) 2 ) ^ ] 2 ( n o t e t h a t a l s o h e f w j J w ^ 3 w a s assumed). T h e r e f o r e || w^ - w^ . ||2 -• 0 as 6 -• 0 , c o n t r a d i c t i n g || - w^ . || >_ g . Hence w^ | [ ] . F i n a l l y , i f w^ = ^ h + P 2 w k >• t h e n w^ = P i a 2 . w j + (P]_ a2 + ^ 2 ^ w k ' T h e r e f o r e e i t h e r {h,w^ .,w^ .} or Ch-j_w^ ,w^ } i s inde p e n d e n t . C o n s i d e r now the t h r e e d i m e n s i o n a l l i n e a r subspace M = [h,w .,w, ] of H ; d e f i n e a C a r t e s i a n c o o r d i n a t e system In M h a v i n g u n i t v e c t o r s x., ,x ,x ( i . e . ( x . , x . ) = 6..) h " w i t h x-, = — — — ' . Then M i s i s o m o r p h i c a l l y i s o m e t r i c t o R under the map I : u = S cc^x^ -• (a-^,a 2,a^) . F o r E = M n S = [u e M | || u |! = 1} we have 40. IE = {(a^,a^,a ) e | E l a j _ | 2 i s bounded} since || • [[ and j| • || are equivalent on M ( f i n i t e dimensional); i n addition, IE i s a quadric surface, since || • || i s from an inner product, so that IE i s an e l l i p s o i d i n Ir . By the choice of w, ,w. • . - . . . . K J . this e l l i p s o i d s a t i s f i e s Lemma 2.6, so we conclude that there i s an x e S with || x || < J\i^ , contradicting the d e f i n i t i o n of [_t^  . Therefore {w^} again has a convergent subsequence. The following lemma i s v a l i d f o r either r e a l or complex Hilbert spaces. Lemma 2.8 Suppose {c^}, {v^}, v^ e S , are such that || v x ||2 = i n f || u ||2 [u e S] = a 1 , |j v. ||2 = p ) i n f || u | r [u e S n V±_°_] = c i ( i = 2,...,N) , where V. 1 = [v 1 ,. . .,v. 1.] . Then we have (v,v. ) = cr. (v,v. ) f o r any v e H Q , i = 1,2,...,N . Also i f any v e S has II v J)2 - a± , (1 1 i 1 N) , then || x ||2 = o± f o r any x e [v,v i] fl S . Proof: (Induction on N). Let II v n | [ 2 = o\ and v e H . 1- • 1. o be fixed with (v,v n) = 0 . Then for u e S , . .x 1 o. ' || u |J2 - o 1l| u ||2 >_ 0 , with equality f o r u = v 1 . Hence the same holds for any u e H Q . Therefore f ( ^ ) = || v 1 + £v|| C r 1 J| + €v 1  2 >_ 0 with minimum f ( o ) = 0 . Hence 2(v,v-j_) = f'(0)^ = 0 . We have shown (v,v 1) = a 2 _ ( V J V 1 ) 0 i f ( v , v 1 ) Q = 0 . But now, f o r any v e H Q the same i s true, because writing v.= av^ + g , with a = (v,v^) , we have 41. (g,v^) = 0 , so t h a t ( v , v ^ ) = acr^ = a]_ ( v .J v ] _ ) 0 • ^° f i n i s h the case N = 1 , . i f v e S a l s o has || v ||2 - ^ e n ^he above e q u a l i t y shows || av + Sv-^ ||2 = c^|| av + Bv^ ||2 ,. wh i c h eq u a l s f o r av + Bv-j^ e S . Suppose now t h a t t he theorem i s t r u e f o r N _£ n - 1 , and || v n ||2 = CTn • Then the above argument, r e p l a c i n g S by s ' '= S n [ v 1 , . . . J V ^ ] 1 ° , g i v e s ( v , v n ) = 0 f o r any v e [ v - ^ . . . ^ v n _ i ^ a s a t i s f y i n g ( v ^ v n ) Q = 0 • B u t then t h i s i s a l s o t r u e f o r any v e H 0 : i f v e H Q and ( v , v n ) 0 = 0 > n - 1 ' ' j_ then by w r i t i n g v = £ a j _ v j _ + w > where w e . [v.^ . . . , v ^ ^ } °, we see (w,v n) = 0 and ( v n > Y 1 ) = 0 ( i = l , . . . , n - l ) , so t h a t ( v ^ v n ) = s a j _ ( v j _ ^ v ) + ( w , v n ) = 0 • T h i s i s the con-c l u s i o n , s i m i l a r t o the one f o r the case N = 1 , whi c h enabled us t o show (v,v n) = a n ( v * v n ) 0 f o r any v e H Q . The r e s t o f the d e t a i l s a r e the same as f o r t h a t c a s e . Lemma 2 . 9 F o r n >^  2 , the e x i s t e n c e o f n l i n e a r l y inde-pendent v e c t o r s u^,...,u n w i t h || u-^  ]|= = = i n f • || u || 2 [u e S] and [| u ± ||2 - u ± = i n f || u || 2 [u e- S fl -~^ ~f- l ( 1. = ;->. n ) an rt i\i . . \:. . , . . . U i ° 1 ] i  2,3,...,n) d ( u ± , u )Q = 6 .  i m p l i e s W + k = Xn+l+k = s u p i n f I' u l ! 2 [ g i ^ - - ^ e u n " u e S fV ' ir^ n [g-j_, . . . , g k ] 1 0 ] - f o r k = 1 , 2 , . here the g^'s a r e assumed t o be l i n e a r l y • i n d e p e n d e n t and as 4 2 . u s u a l U n = [ u ^ , . . . ) ^ n ] P r o o f : F i x k >_ 1 , n >_ 2 . By d e f i n i t i o n o f l-^+i+k ; w e see t h a t i t s u f f i c e s t o show H ^ J L ^ <. ^ n + i + i c • F o r t n i s ^ l e t f ^ , • • • • ) f n + l c "be f i x e d l i n e a r l y i ndependent v e c t o r s i n H Q . We i n t e n d t o show i n f || u | | 2 [ u e S fl [ f - ^ . . . , f n + k r H 1 X n + l + k * F i r s t , t o f i n d s u i t a b l e g's : " n + 1 ' l e t g n = l a . f . where the a-, ' s a r e chosen t o s a t i s f y the e q u a t i o n s g 1-L 0u- L, . . . ,u ; suppose i s the l a r g e s t n+ 2 i n t e g e r w i t h CL „ ^ 0 . Next l e t g 0 = £ a 0 . f . , where 5 1 1 = 1 * i A x g 0 ± 0 u 1 , . . . ,u and £ 0 i s maximal w i t h a f ^ 0 . S i m i l a r l y c j . • n d d, n+k . ' d e f i n e ' g^,...,g k = Y,^ a k , i f i w h e r e s k ^ u l ' ' ' * , U n a n d . . . ( I, i s maximal w i t h a, . ^ 0 . N o t i c e t h a t the g. ' s a r e • iC K., k independent. We a r e now ready t o show i n f H u || 2 [ u e S n [f±, . . • , f n + 1 J ^ 1 i n f || u |.|2[u e S n I T ^ 0 n [ g x , • - • ,gk]*11 : l e t u e S n n [ g x , . . . ,g k] X° • L e t n u' =• £ a.u. + a _ u , where the a. 's a r e dete r m i n e d by 1. l l n + 1 J l J || u 7 || Q = 1 and u'-Uf^ f o r i = 1 , 2 , . . . , n + k except f o r i = . Then i n f a c t u' i s o r t h o g o n a l t o f f . and || u' ||2 < || u ||2 : f i r s t , 43.' J^2_ •'••••» ^ _ 1 , . . . ,t k-1 s i n c e g k_L ou^ ., . . . ,un.,u . T h i s i n t u r n i s e q u a l t o s i n c e u' i s o r t h o g o n a l t o t h e s e f - j ' s • I n the same way u' i s o r t h o g o n a l t o the r e m a i n i n g f . ' s ( t a k e n i n the o r d e r f f ) . J V l ll F i n a l l y we show || u ' | [ 2 <_ || u ||2 : I! u || 2 = J I a ± | 2 II u ± ||2 + | a n + 1 | 2 || u ||2 + c r o s s - t e r m s . However the c r o s s - t e r m s o f type c(u.,u.) = 0" because x j c(u.',u.) = c u. (u. ,u.) from Lemma 2 . 8 ; the r e m a i n i n g c r o s s -terms a r e o f type c ( u ^ u ) and th e s e e q u a l c U ^ ( U ^ , U ) Q = 0 a g a i n by Lemma 2 . 8 and the c h o i c e of i i . T h e r e f o r e II u' ||2 < £ \a±\\ + | a n + 1 | 2 | | u | | 2 < .||u-||2 T l a . J 2 = | | u | | 2 s i n c e H ± < H n < II u ||2 (u e I f ^ n S) . 44. P r o o f o f Theorem 2 . 5 - ( I n d u c t i o n ) . Suppose k^ < °= , so t h a t > = l i k i _ x = ... = ^ . F i n d f ^ - . . , ^ e H Q such t h a t ( 2 ) i n f !| -UL (12 [u e S n [ f ^ . . . , ^ J 1 " ] > u S i n c e u k - , f o r any v e c t o r s h ^ , . . . , ] ^ i n H Q we have u 1 <_ i n f |J u | | 2 [ u e S fl [ h ^ . . .• , h k ' ^ ^ J < u, T h e r e f o r e the f . 's a r e n e c e s s a r i l y i n d e p e n d e n t , ^ - 1 so we may w r i t e f v = £ cs.f. + h , h £ 0 , where K l i = l 1 1 h / X« h U » • • • , f k - _! J put . g = — - n — . . F o r H = [ f ,. . . , f ahd s' = S n E'o , we have by ( 2 ) t h a t i n f || u | | 2 [ u e S* n [g] 1*] > u x = i n f || u | | 2 [ u e s/.] . A p p l y i n g Lemma 2 . 7 w i t h H^,S' , we. f i n d u^ e S* , || u^ ||2.= u.^ Repeat the above p r o c e d u r e , u s i n g f, n i n s t e a d o f f, , t o 1 " 1 f i n d u 2 e S* = S n [ f 1 , - - - . f k i _ 2 ^ f k 1 ] " L ° 3 " U 2 " 2 = ' ^ 1 * I f 1 2 u l = U 2 ' t h e n u i e t ^ j - - - ^ ^ 1 ° s o t h a t || •u 1 || > u-j, , a c o n t r a d i c t i o n . I n t h e same manner we o b t a i n l i n e a r l y i n d e p e n d e n t v e c t o r s u^,. . . , u k w i t h u^ e S , || u^ || = p.^  . By 45. Lemma 2.8 we may orthogonalize these vectors (by the Gram-Schmidt procedure), again c a l l i n g them u, ,.. . ,u, and 1 v e [ u ^ . . . , ^ ] n S implies || v || = |i 1 . 2 Suppose now v e S , || v [j = |X-^  and v <± [u-,,...,u. ] ; ¥e intend to f i n d a contradiction. Find 1 *1. x e ' [v,u v ] n S such that (x,u. ) = 0 ; then || x || = \i k 2 - l by Lemma 2.8. Then f o r y = £, a.u. + £3x , where a. ,(3 are i-1 1 1 1 determined by || y ||q = 1 , A ^ v ^ ^ ; we have ( y * ^ ) D 2 = 0 so that || y || >_ U k + 1 > U k . However by Lemma 2.8, 2 2 a simple c a l c u l a t i o n shows || y [| = u., (j y || = u, = u, , a JL O _L contradiction. Thus we have that u,,...,u are eigenvalues (Lemma 2.8 again) with orthogonal eigenvectors u-, ,.. . ,u, JL j~ s a t i s f y i n g the f i r s t parts of the theorem. It.remains to show that u ± = i n f J| u ||2 [u e S fl ( i = 2,...,^+!) . This i s clear for i = 2,...,^ , so consider i =.^+1 . Let x k n + l = i n f 'I u " 2 C u e S 0 Uic°-I ' B Y L E M M A 2-9 (n.= k - ^ l , k = l) , P k i + 1 = sup i n f || u ||2 [g £ H Q n U k L-_ 1 I ' u e S n H [g]"1"0] . Therefore i t s u f f i c e s to show 1 ^ +]_ 1 ^ + i • 46. Let g e H Q n U ^ _ 1 , u € S n , u - L u ^ . Then f o r u' = au + S u ^ , where ( u',g) Q = 0 , J| u ' | | 2 = 1 , .we have || u ' ||2 = |a| 2 || u ||2 + || u, ||2 < || u ||2 , where the 1' equality i s true because the cross-terra i s zero by. Lemma 2 . 8 , 2 and the ineq u a l i t y i s true because || u, || = u v = *1 K l i n f || v ||2 [v e S n U k ° _ x ] £ || u ||2 . • This shows p k + ^ <_ af t e r taking i n f and sup. . To.complete the proof of the theorem, assume i t true for p-, ,. . . ,p -, and suppose k < <» . I f p = p 3 then the inductive assumption f o r p - 1 i s the proof. Suppose Pp > n p_ 1 . Let = [u 1,...,u p_ 1] X° , S' = S fl r . By 2 by Lemma 2 . 9 the remaining higher eigenvalues are given by = S U P i n f II u II 2 [ g ^ — o g ^ 6 | u e U' n [ g 1 , . . . ,gk]"L°] the inductive assumption we have p. = i n f || u || [u e S'] and xp+k > = case u-^ k.^  . (k = 1 , 2 , . . . ) . Since k < » , this i s now reduced to the Lemma 2 . 1 0 Let = i n f B[u,u] [u e S] be. assumed, at v^ and a n = i n f B[u,u] [u e S D V ^ - j J be assumed at v^ .. (n = 2 , 3 , . . ' . ) . Then a n = for a l l n . Remark By Lemma 2 . 8 , the °" n , s a r e then eigenvalues of B Proof: To show a n > p n , l e t x 1,...,x n_ 1 be given i n H Q 47 n For v = £ c^v , determine c^'s °y the equations ( v ^ x j _ ) 0 = 1 ( i = l , . . . , n - l ) and | | v f | | o = l . Then B[v,v] < a j l v o We conclude the chapter with the following Lemma 2.11 I f = B t u ^ u ^ ' for some u i e H D ( i = 1 J - - • where (u-,u.) = 6 . , then the supremum defining u. can be taken over the larger family o f a l l i - l dimensional subspaces of L 2(G) ( I = 2,...,n) . Proof: Let 2 <_ k <_ n . It i s enough to show >_ sup i n f B[u,u] [P e P k_ x 3 | u e S fl PX<>] . Fix F 1 J - - - > F L C _ 1 i n L^(G) and set u = E c u . : determine the c. 's by the 2 V l l ' x J k-1 equations ( u ^ j _ ) 0 ~ 0 ( i = l j . - - . j k - l ) and the normal-11 • I r T & 2 - 2 i z a t i o n || u || Q = 1 . Then BLu^uJ = E c i n i < p k E c i = . Remark Lemmas 2.10 and 2.11 are also v a l i d under the as sump tions that the spaces and forms are i n the complex setting. This w i l l be required for l a t e r r e s u l t s . CHAPTER THREE DEPENDENCE OF EIGENVALUES ON THE DOMAIN For s e l f - a d j o i n t second order d i f f e r e n t i a l operators L , defined over bounded domains G , i t i s well known that under s l i g h t assumptions on the c o e f f i c i e n t s , the eigenvalues depend on the domain i n a monotonic manner: generally speaking G <= G* implies X^ >_ X? for corresponding eigenvalues; i n fact s t r i c t i n c l u s i o n implies s t r i c t i n e q u a l i t y (cf. [13], [ 2 0 ] , [ 2 2 ] ) . We would l i k e to generalize this relationship to the eigenvalues of b i l i n e a r forms, and some of the singular operators mentioned i n Chapter One. A. Weak Monotonicity Property f o r Forms Let B be a b i l i n e a r form defined on a space H(G) C: Lg(G) that i s the completion of C Q ( G ) with respect to some metric. Theorem 3.1 Suppose the eigenvalues of B s a t i s f y = min B[u,u] [u e S] , u n = min B[u,u] [u e S n U^l ]_3 (n. = 2,3,...), where as usual S = {u e H | || u || = 1 } , -u4- = {u e H | o j (u,u^) = 0 ( i = 1 , . . . , j ) , u^'s being the eigenvectors corres-ponding to the u^'s} . I f , u? e H(G*) are the eigen-values, and eigenvectors of the same problem posed over the (open) domain G* , then G* £ G implies u <_ u* . 4 9 -P r o o f : Analogous to [ 1 3 ] , an a r t i f i c i a l I n t e r m e d i a t e p r o b l e m i s s e t up. F o r t h i s - w e need to extend f u n c t i o n s of H(G*) to H(G) by z e r o e x t e n s i o n ; i . e . i f f e H(G*) , t h e n f i n d 'f e C™(G*) w i t h f n -* f . S i n c e each . f n has compact s u p p o r t , i t v a n i s h e s i n a neighbourhood o f 3G* , so may be extended by z e r o and s t i l l has r" n e C Q ( g ) • B u t the n t h e s e extended t e s t f u n c t i o n s a p p r o x i m a t e the z e r o - e x t e n s i o n o f f , f" , i n H(G) , so r" e H(G) . . I f n = 1 ,, the n c l e a r l y ^ — ^n " I f n > 1 3 then c o n s i d e r u** = i n f B[u,u] [ u 6 S* fl " ^ n - l ^ 3 where u^ e H(G) a r e as above. Then u** >_ because H(G*) £ H(G) by e x t e n s i o n s . To see p* >_ u** , we o n l y have to n o t e t h a t U£ = sup i n f B[u,u] [f13 . . . , f n _ ] e L 2 ( G ) | u . e S*fl [ f ^ , . .., f ^ by the p r o o f s o f Lemmas 2.10 and 2.11. For t h e above e i g e n v a l u e p r o b l e m , i f B[u,v] = (vu,vv) + ( V U , V ) q + (u,v) , d e f i n e d on H = W ^ 2 ( G , V ) ( o r wJ j 2(G) i f V = 0) , and L = -A + V + I , d e f i n e d on £(L) = H fl {u e L 2 ( G ) | Lu e L 2 ( G ) } , so t h a t B[u,v] =" ( L U , V ) Q f o r u e £ (L) , V e H (see Chapter One),, t h e n the e i g e n - ' f u n c t i o n s o f B ' c o u l d be thought o f as s a t i s f y i n g a g e n e r a l i z e d . D i r i c h l e t c o n d i t i o n , i . e . u = 0 on 3G i n some sense. I n g e n e r a l ( c f . [21]) i f H i s a v e c t o r space 5 0 . s a t i s f y i n g W ^ 2 ( G ) c ' H £.W1>2{G) , w i t h H c l o s e d i n ¥ 1 > 2 ( G ) , n and B[u,v] E f a . .(x) D .u D.v dx + f V ( x ) u v d x f o r i , j = l G • J 1 J G . u,v e H (a. .,V r e a l v a l u e d ) , w i t h B c o e r c i v e o v e r W^ j 2(G) i n L 2 ( G ) , then one c o u l d c o n s i d e r the o p e r a t o r n 3 , , N a Lu =.- Z v ^ - ( a . . ( x ) -K— u) + V ( x ) u w i t h domain ^ ( L ) = i , d = l S x i . 1 J . • d x j • • {u e H | Lu e L„(G) ; J vLu dx = B[u,v] f o r any v e H} . A G f o r m a l , u s e o f Green's f o r m u l a would g i v e J v ~ da = 0 3G L f o r u e £(L) , where = £ a.^  . c o s ( n , x ^ ) , n b e i n g a L u n i t v e c t o r i n the d i r e c t i o n o f t h e o u t e r n o r m a l . T h e r e f o r e one c o u l d a l s o c o n s i d e r the f o l l o w i n g " l i m i t " p roblems: ( i ) i f H = ¥ l j , 2 ( G ) , then u e Jb(L) would f o r m a l l y s a t i s f y u = 0 on 3G ; i . e . a g e n e r a l i z e d Neumann c o n d i t i o n ; L ( i i ) i f 3G i s s u f f i c i e n t l y smooth, 3-^ G i s a p a r t o f 3G , and H = [u e W 1 ; , 2(G) | u = 0 on 3 ^ } , then u e .fr(L) would f o r m a l l y s a t i s f y u = 0 on 3-, G , u = 0 on 3G - 3-, G : 1 3 v L 1 i . e . a g e n e r a l i z e d mixed c o n d i t i o n . There i s a r e l a t i o n s h i p between the e i g e n v a l u e s o f a domain and subdomain f o r the g e n e r a l i z e d N.P. mentioned above. I f G = Gj U Gg u F , where G^ are. open, G 1 fl G g = <f) , and F c G i s the boundary s e p a r a t i n g them, th e n ¥ 1* 2(G^ U G p) = ¥ 1 > 2 ( G 1 ) © ¥ 1 , 2 ( G 2 ) and f o r any u e ¥ l j 2(G- L I . ) G 2 ) , we have 5 1 . B[u,u] = B 1 [ u 1 , u 1 ] +'B 2[u 2,u 2] where u .= ^  + u g e ¥ 5 (G^) and B^ i s the sa m e . e x p r e s s i o n as B , i n t e -g r a t e d over G^ . Theorem 3 . 2 . Suppose the e i g e n v a l u e s o f B s a t i s f y ^ = min B[u,u] [u e S] , u n = min B[u,u-] [u e S fl U ^ - j J (n = 2 , 3 , • • - ) where ' S = {u e ¥ 1 > 2 ( G ) | [| U ||Q = 1 } , U j° = [u e W 1'^) | ( u , u ± ) o = 0 ( i = 1 , 2 , . . . , j ) } , and where 1 2 i the e i g e n f u n c t i o n s u^ e ¥ > (G) ( r e s p e c t i v e l y u n a r e the e i g e n f u n c t i o n s f o r B^ s a t i s f y i n g c o n d i t i o n s s i m i l a r t o t h e above ones, and u ^ e ¥ 1 , 2 ( G ± ) ( i = 1 , 2 ) ' ) . Then ^ > l n , where ( X n ) i s the sequence o f u^'s , w r i t t e n i n i n c r e a s i n g o r d e r . P r o o f : L e t v n denote the e i g e n f u n c t i o n s f o r X n , extended by z e r o t o G^ U G 2 , and f o r any u e Lg(G) , l e t u | G = U|Q. = u 2 . ¥e u t i l i z e t he f a c t t h a t e i g e n v a l u e s f o r non-connected domains a r e p r e c i s e l y the e i g e n v a l u e s o f a l l components. F i r s t we show >_ X^ (we use a p r o o f t h a t c a r r i e s o v e r t o h i g h e r n ). For t h i s , u.^  = i n f B[u,u] [u e ¥ 1 * 2 ( G ) , || u ||D = 1 ] > i n f B[u,u] [u e L g ( G ) , || u ||q = 1 , e ¥ l j 2 ( G 1 ) , r l 2 1 2 u 2 e W*^  ( G 2 ^ - l = m i n £ I-1! * ^ i ^  = ^ i • ^° s e e t h e s e c ° n d l a s t i n e q u a l i t y , l e t u s a t i s f y t he f o u r c o n d i t i o n s . I f u^ = 0 o r u 2 = 0 ., th e n c l e a r l y B[u,u] >_ . I f u^ ^  0 , u g ^ 0 , 5 2 . t h e n B [ u , u ] = s| | u. | | 2 B [ 1 , 1 ] > ^ . T h e r e f o r e ' " u i " o " u i " o i n f B [ u , u ] [u e W l i 2 ( G 1 u G 2 ) ; || u || Q = l ] >. ^  '. The o p p o s i t e i n e q u a l i t y i s o b v i o u s . To s e e u. > X i n g e n e r a l , we n o t e f i r s t t h a t U n = sup i n f B [ u , u ] [ P ^ c L 2 ( G ) | u € W 1 ' 2 ^ ) , |[ u || Q = 1 , u i 0 P n ^] where ? n _ i I s a r i y n _ l d i m e n s i o n a l s u b s p a c e o f L 2 ( G ) . ( c f . p r o o f o f Lemma 2 . 1 1 ) . T h e r e f o r e u n >_ i n f B [ u , u ] [u e ^ ^ ( G ) , || u || Q = 1 , u i , v . ( i = 1 , . . . , n - l >_ i n f B [ u , u ] [u e L g ( G ) , " u ± e ¥ l j 2 ( G i ) ( i = 1 , 2 ) / || u || Q = 1 , u l 0 v ^ ( i = 1 , . . . , n - l ) ] = X n , where t h e l a s t i n e q u a l i t y i s done e x a c t l y as f o r t h e c a s e n = 1 . B. S t r o n g M o n o t o n i c i t y P r o p e r t y f o r Forms. U n d e r a few a d d i t i o n a l r e s t r a i n t s , t h e e i g e n v a l u e s o f forms c o r r e s p o n d i n g t o a G.D.P. s a t i s f y u * > [i i f • G* 5 G . F o r t h i s we r e q u i r e a n a p p l i c a t i o n o f t h e u n i q u e c o n t i n u a t i o n p r i n c i p l e f o r s o l u t i o n s o f e l l i p t i c e q u a t i o n s . n I f L = E D . ( a . .D.) + V i s an e l l i p t i c o p e r a t o r i , j = l - • w i t h a n a l y t i c c o e f f i c i e n t s , t h e n a ny c l a s s i c a l s o l u t i o n o f L u = f t h a t v a n i s h e s i n an open s e t o f G v a n i s h e s i d e n t i c a l l y i n G , p r o v i d e d f i s a l s o a n a l y t i c . U n d e r c e r t a i n l e s s r e s t r i c t i v e c o n d i t i o n s c l a s s i c a l s o l u t i o n s r e t a i n t h i s c o n -t i n u a t i o n p r o p e r t y ( c f . [7] and t h e b i b l i o g r a p h y t h e r e ) . 5 3 . Since the conditions for i n t e r i o r r e g u l a r i t y of weak solutions are well known, we have the following Lemma 5.5. Let G be a bounded open set i n E 2 1 . I f the co e f f i c i e n t s of L-j_ = L - X are smooth enough' so that L^u = 0 has a weak solution ( i . e . (u,L*f) = 0 f o r any f e C Q ( G ) where L£ i s the formal adjoint of L^) and i f , i n addition, ( i ) L i s e l l i p t i c i n G and ( i i ) the c o e f f i c i e n t s of L are Holder continuous, then the solution u e ¥^" , 2(G) i s i n fact a c l a s s i c a l solution (a.e.). I f u = 0 i n an open neighbourhood N c G , then u = 0 i n G . Proof: By the f i r s t two conditions, u i s a c l a s s i c a l solution almost everywhere (cf. [ 7 ] ) , and the t h i r d condition then f u l f i l l s the requirements f o r L^ to have the. unique continuation property (cf. [ 2 7 ] , [ 2 9 ] ) . We now proceed with the strong monotonic property. Theorem 5.4. Let B[-,-] be a' b i l i n e a r form, coercive over c"(G) i n L 2 ( G ) . Let H denote the completion of C~(G) ( i d e n t i f i e d i n L^(G)) with respect to the metric from B , and denote the extension of B to H again by B . Let L be an operator s a t i s f y i n g the conditions of the above lemma and having. L(u,v) = B[u,v] -for any u,v e C™(G) . Let ^1 — ^2 — "'" ^ e the eigenvalues of B , given by p. = min B[u,u] [ u e S D u i ' 0 1 ] , as i n Theorem 3 . 1 , with 54. eigenvectors e H(G) ; r e s p e c t i v e l y u£,u* e H(G*) , where G* Is an open sub-domain of G such that G - G* contains an open set. Then > i s true f o r the corresponding n ^ eigenvalues provided B has an eigenvalue over G l a r g e r than ^n • Proof: The proof proceeds by c o n t r a d i c t i o n . Suppose G* and G s a t i s f y the hypotheses and yet n n = u* . By assumption, there' e x i s t s k w i t h u^ . > u n . We then p a r t i t i o n G i n t o k s e t s , G* = G-^  c Gg ,.. c G^ . = G , the only requirements being that G^ be.open and G - G^  c o n t a i n an open set ( i = l , . . . , k - l ) . By the weak monotonic pr o p e r t y (Theorem 3 . l ) , 1 2 k i " th we have 1^ * = H n >. U n >. • • • >. V-n = U n , where u n i s the n eigenvalue, of B f o r the problem posed i n H(G i) . Denote the corresponding eigenfunctions by v^ ( i = l , . . . , k ) . Extend v^ to G by zero extension, so that v^ e H(G) , || v^. ||o = 1 . The f o l l o w i n g argument shows that { } ^ are l i n e a r l y inde-k k - 1 a. pendent: i f E a v. = 0 and a, ^ 0 say, then v. • = - S — 1 x 1 K K -, a, 1 k i s i d e n t i c a l l y zero on G - Gk_-j_ /• Since v^ € L 2(G) , we can f i n d C l a r g e enough so that f |v,| dx < 1 , where G-B (C^) B ( C , i ) = (x e G | |x| 2 < Q , d i s t (x,H> ) > i } , and such that B ( C ^ ) i n t e r s e c t s G - G k_ 1 i n an open set. Since v f c i s a weak s o l u t i o n o f N (L - p^) u = 0 , then by the unique c o n t i n u a t i o n property (Lemma. 3 - 3 ) , v f e s 0 on B ( C ^ ) . Hence || ||Q < 1 , a c o n t r a d i c t i o n . Thus a, = 0 : s i m i l a r l y the other a.'s K X V. k 1 55. are zero. Now l e t u, ,.. . ,VL n be the eigenfunctions i n H(G-) corresponding to u, ,...,u, , . For f = E c.v. , the homo-geneous system of equations (f,u^.) G = 0 ( j = 1,.'. .,k-l) has a n o n - t r i v i a l solution with || f || = 1 , by the l i n e a r inde-pendence of the v i's . F i n a l l y , u ^ . = min B[u,u][u e S f l ] iBCf,f] = i,ii0i°j Btvi,vJ]=^ i,Lcic^Vi*vJ?°= p.n || f Ii2 = H n . This contradicts the assumption > u n , so that | i * > H n . C. Summary By the results of Chapter Two and the above, we may conclude the following Theorem 5-5. Let B be a coercive form over a r e a l space H with B[u,v] = (Lu,v) for any u,v e C™('G) cz H , where L i s a s e l f - a d j o i n t operator s a t i s f y i n g Lemma 5.5. I f the numbers , defined i n Theorem 2.4, s a t i s f y ^ < p.^  , then they are eigenvalues of B , hence of L , enjoying the properties of monotonic dependence on the domain G as given by Theorems 3 .1 , 3 .2 , 3.4. 56. Remark The eigenvalues of the. semi-discrete operators, i n Chapter One that were given by \ ± = i n f B[u,u] [u e H q fl U^ L^  | | u | | 0 = l ] (1=1,2,...) (cf. Theorems 1.2, 1.6, Lemma 1.5) also s a t i s f y the above monotonic relationships, since X 1 = wu ( i = 1,2,...) 'by Lemma 2.8 and a l l the relationships were proved for {u.3 . CHAPTER FOUR APPLICATIONS OF MONOTONOCITY A. Symmetry In the c l a s s i c a l theory of many eigenvalue problems, i t i s observed, that a symmetry i n the underlying domain Induces a symmetry i n the eigenfunctions (cf. [17]). Required i n the proof i s the strong monotonicity dependence of the eigenvalues on the domain. We use the results of the f i r s t three chapters to prove this symmetry property f o r the eigenvalues of -A over an unbounded domain G that has a symmetry. Theorem 4.1. . Let G be an unbounded domain with a symmetry about a hyperplane, denoted z = k , and l e t L denote the F.E. of -A over C~(G) . I f X-j_ <_ X 2 <_ ... are the eigenvalues 1 2 of L , with corresponding eigenfunctions u^ e ¥ q 5 (G) , then the f i r s t eigenfunction, u^ , has symmetry (a.e.) about z = k . If i n addition X^ i s a simple eigenvalue, then u^ has symmetry (a.e.) about z = k i n the more general sense that there exists a constant, c. , with u'. (z) = e.u. (z, ) where . z, i s the r e f l e c t i o n of the point z with respect to z = k '. The proof requires the following Lemma 4.2. Let G be bounded (with BG smooth). I f u e C°°(G) and u(x) = 0 on 3G , then f o r any € > 0 , there 58. exists f e C Q(G) with j| f-a \\1 2 < e . Proof: ' By a proof i n M i k l i n [25], there exists v e C^(G-) with v vanishing i n a boundary layer of G and || v-u | j ^ 2 < G/2 . By regu l a r i z a t i o n (cf. [ 5 ] ) , there exists a > 0 such that the " m o l l i f i e r " J g ( v ) e C~(G) and J 0 ( v ) - v as a - 0 i n W^ 2(G) . For small enough a , |[ J (v) - u |L 0 < € so take f = J (v)•. Proof of Theorem 4 . 1 : Let B[u,v] denote the b i l i n e a r form B[u,v] = (vu,vv) Q f o r u,v e W^ , 2(G) ,. so that L and B have the same eigenvalues and eigenfunctions (Theorem 2 . 5 ) . Further, suppose G = Q 1 U Q 2 U (z=k) , where Q 2 i s the r e f l e c t i o n of with respect to the hyperplane z1 = k . Also denote by z^ . the r e f l e c t i o n of z with respect to z = k . F i n a l l y l e t z) = u-^z) - ^(z^.) be defined f o r any z e Q-^  . We show u^(z) = 0 a.e. F i r s t u x ( z ) e W^ 2(Q 1) : l e t £ > 0 and f i n d f e C*(G) such that || f - VL1 \\1 2 < € . Define f ( z ) = f(z) - f ( z k ) for z e Q1 . By Lemma 4 . 2 , there exists * e C Q ( Q I ) w l t h II * " f " l , 2 , Q 1 ' < € ' B u t t h e n || ii; - U-L I I 1 > 2 J Q < 2 € - Next, B l u ^ f ] = X 1 ( t i 1 , f ) for any f € W^' 2^) : l e t f e C"(Q X) and define f k ( z f c ) = f ( z ) for any z e Q-^  Then f ^ e C o ^ 2 ^ a n d w e h a v e 59. B t u ^ f ] = J u 1 ( z ) ( - A f ( z ) ) d z = J u 1 ( z ) ( - A f ( z ) ) d z -Q l ' " Q l ' - J u 1 ( z k ) ( - A f k ( z k ) d z = X l t f u x f d z - u 1 ( z ) f k ( z ) d z Q 2 Ql Q 2 = X 1 ( u 1 , f ) • . . 1 2 By continuity, this i s also true for f e W 5 (Q^) . Therefore we have u^ = 0 ( i n ^(Q-^)) or else i s an eigenvalue for B over . By the strong mono-t o n i c i t y property (cf. the remark following Theorem 3-5), the l a t t e r cannot happen. Thus symmetry i s shown. For the case of simple eigenvalues, suppose X n i s a simple eigenvalue and'again define ^ ( z ) = u n ( z ) ~ u n ^ z k ^ fo r z e Q-^  • By the same argument as above, either u n = 0 or X n i s an eigenvalue f o r B over Q-^  , with eigenfunction 1 2 ~ -u^  e W.o* (Q-j_) • I f the l a t t e r i s true, l e t ^ n ( z ) be defined. over G by u n ( z ) = % ( z ) i f z e ; = - u n ( z k ) i f z e Q2: Then c l e a r l y u n e w^'2(G) and we have B[u^,f] = ^ ( u ^ f ) for any f 6 wJ j 2(G) : l e t f e C*(G) and f k ( z ) = f ( z k ) for any z € G . Then B[\f , f] = P u ( z ) ( - A f ( z ) ) d z -. Qx - • • J u n(z k).(-Af(z)dz = J* u n ( z ) ( - A ( f ( z ) - f k ( z ) ) d z . By Lemma Q 2 . Q l ' • 4 . 2 , f - f*^ . J Q € ¥ ^ , 2 ( Q ^ ) . . Since u n i s an eigenfunction, 6 0 . the l a s t expression Is equal to X n ( u n ( z ) , f ( z ) - f^(z))^ = continuity this i s also true f o r f e ¥^ J Hence again u^ = 0 or S n i s an eigenfunction f o r X , i n which case u* = c u . I f u* = 0 . then f o r almost A n ' n n n 3 a l l z € we have u n ( z ) = 0 ; i . e . u n i s symmetric. I f u^ = c u n , then f o r almost' a l l :z e Q-^  , ^ n ( z ) = c u n ( z ) o r (c+1) u n ( z ) = c n u n ( z ) - u n ( z k ) . B. ' Miscellaneous For a coercive form B , with domain H , l e t be defined as i n Chapter Two, with (_t < u . I f H i s ¥^* 2(G) (or some completion of C^(G)) , we have seen that u-^,...,u-n are eigenvalues corresponding to a D i r i c h l e t 1 2 problem; i f H i s ¥ ' (G) , they correspond to a Neumann or mixed problem. Since ¥ 1 , 2 ( G ) c ¥ 1 , 2 ( G ) , the eigenvalues of the. Neumann problem w i l l always be smaller than or equal to the corresponding eigenvalues f o r the same D i r i c h l e t problem. I f one i s interested i n the growth of such eigenvalues, (cf. [12] and the references given there),, then a "trace function", N(u) = S 1 , i s defined, where u. are the eigenvalues i n ' ^ X 3 question. Let N^(u) denote such a function f o r the D i r i c h l e t 61. problem over a domain G' , and. N^( p.) be the same f o r the same Neumann problem. Then by the results of Theorems 3 . 1 , 3-2 , i f n G = IJ G. , where G. are non-overlapping sub domains, we have . 1 1 W N°'(u) < N*J(u) < An) < S Np ( u ) • 1=1 u i . ~ u u. - 1=1.^1 • These results are of course well-known f o r regular eigenvalue problems [ 13] . 62. BIBLIOGRAPHY [ l ] R. A. Adams, Compact S o b o l e v I m b e d d i n g s f o r U n b o u n d e d Domains w i t h D i s c r e t e B o u n d a r i e s , J . o f M a t h . A n a l , and A p p l . , V o l . 24 No. 2 (1968) . [2 ] R. A. Adams, The R e l l i c h - K o n d r a c h o v T h e o r e m f o r U n b o u n d e d D o m a i n s , A r c h i v e R a t . Mech. A n a l . , 29 (1968) . [ 3 ] R- A. Adams, C a p a c i t y a n d Compact I m b e d d i n g s , J . M a t h . Mech. ( t o a p p e a r ) . [ 4 ] R. A. 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T r a n s l a t i o n s , Ser ies 2 , V o l . 13 ( I 9 6 0 ) . [17] J . Hersch, E r w e i t e r t e Symmetr ieeigenschaften von L<5sungen gewisser. l i n e a r e r Rand - und Eigenwert probleme, J . Reine Angew. M a t h . , 218 ( 1 9 6 5 ) . [ l 8 ] M. R. Hestenes, A p p l i c a t i o n s o f the Theory o f Quadrat ic Forms i n H i l b e r t Spaces to the Calculus o f V a r i a t i o n s , P a c i f i c J . M a t h . , V o l . 1 ( 1 9 5 1 ) . [19] T. Ka to , P e r t u r b a t i o n Theory f o r .L inear Opera to rs , S p r i n g e r - V e r l a g , 1 9 6 6 . [20] R. L e i s , Zur Monotonie der Eigenwerte s e l b s t a d j u n g i e r t e r e l l i p t i s c h e r D i f f e r e n t i a l - g l e i c h u n g e n , Math. Z , . 96 ( 1 9 6 7 ) . [21] J . L. L i o n s , Equat ions D i f f e r e n t i e l l e s Opera t ionne l les et  Probiernes aux L i m i t e s , S p r i n g e r - V e r l a g ( 1 9 6 1 ) . [22] Yu. I . 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Math. S o c , Vol. 95 ( i 9 6 0 ) . [50] F. Riesz,. B. Sz-Nagy, Functional Analysis, Ungar, 1955. [ 3 1 ] W. Stenger, On the V a r i a t i o n a l P r i n c i p l e s f o r Eigenvalues for a Class of Unbounded Operators, J. Math.Mech. Vol. 17, No. 7 (1968) . 

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