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On Green's function for the Laplace operator in an unbounded domain. Hewgill, Denton Elwood 1966

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ON GREEN'S FUNCTION FOR THE LAPLACE OPERATOR IN AN UNBOUNDED DOMAIN by DENTON ELWOOD HEWGILL B.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1963. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mathematics. We accept t h i s t h e s i s as conforming t o the req u i r e d standard THE UNIVERSITY OP BRITISH COLUMBIA J u l y , 1966. In presenting t h i s thesis 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 University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission-for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain shall, not be allowed without my wri t t e n permission. Department of The Uni v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada i i Abstract Supervisor 1 ?r; Golin W. Clark. This thesis Investigates the Green's functions f o r the operator T defined by * D {f e L 2 ( E ) | A f € L 2 ( E ) } Tt :M -Af f o r f € o§(T). Here H^(E) i s a standard Sobolev space, A i s the,Laplaeian> and E i s a domain i n which i s taken to.be "quasi-bounded". In p a r t i c u l a r we assume that E l i e s i n the half-space x 1 > 0 arid i s bounded by the surface obtained by r o t a t i n g ep(x^) about the -axis, where cp i s continuous, M xj) > 0 and cpke L1(0,+<B) f o r some k > 0. The Green's function G(x,y,\) f o r the operator T + \ i s obtained as the l i m i t of the Green's functions f o r the well lo$oiin\p'roiS.leim;on' tne.' truncated' 'domain E ^ = E f l [ X £ < X], Most of the expected properties of the function are developed including the i i i equality 0 < G ( i , y i x ) <:.K(.p/3) p = |x-y| where K i s the fundamental s i n g u l a r i t y f o r the domain. The; eigenvalues and eigenfunctions are constructed, and i t i s shown that i n *"X n ~* \ i as X » f o r each n, where 1^ n 8 X 1 , 3 \ j a r e the eigenvalues f o r the problem on E x and E respectively. Furthermore, i t i s shown that the eigen-values U n ) are: p o s i t i v e with no f i n i t e l i m i t point, and the corresponding eigenfurictions are complete. . A detailed c a l c u l a t i o n Involving the inequality displayed above showsIthat sbme'itisrate (G^^^) of G(x,y,X) i s a Hllbert-Schmidt kernel. Prom t h i s property of G^ k°^ I t follows that the series S X ^ 2 ^ ' ! i s convergent. Prom the convergence of t h i s series three r e s u l t s are derived. The f i r s t one i s an expansion formula i n terms of the complete set of eigenfunetiohs, and the second i s that some i t e r a t e of the Green's ;function tends to zero on the boundary. The l a s t one Is the construction of the solution H(x>X,f), f o r the boundary value problem AH + XH = f H(x,X,f) ..- 0 as x - 5E f o r a '.Buffleiehtly regular f on E. The f i n a l property of the Green's function, namely, that G(x,yyX) tends to zero on the boundary, i s proved using the f a c t that G^^0^ Is zero on the boundary, and cert a i n i n e q u a i i t i t e s estimating the i t e r a t e s G(x,y, X) i s also shown to be unique. The asymptotic formula a g e n e r a l i z a t i o n of the usual asymptotic formula of Weyl f o r the eigenvalues, f i r s t , given by C. C l a r k , i s derived f o r these q u a s i -bounded domains. F i n a l l y , the u s u a l asymptotic formula due t o Carleman f o r the e i g e n f u n c t i o n s I s shown t o remain v a l i d . Table of Contents I n t r o d u c t i o n C o n s t r u c t i o n of the Green's Function and some of i t s elementary p r o p e r t i e s The Eigenvalues,and E i g e n f u n c t i o n s of the Problem The i t e r a t e s of the 1 Green 'js F u n c t i o n and t h e i r p r o p e r t i e s Boundary behaviour and uniqueness of the Green's FuhctiOh A p p l i c a t i o n s of the Green's Function B i b l i o g r a p h y v l Acknowledgment . The author wishes t o thank h i s a d v i s o r Dr. G o l l n W. C l a r k f o r suggesting the t o p i c of t h i s t h e s i s and f o r - h i s h e l p f u l advice during; I t s p r e p a r a t l o h . He a l s o wishes t o thank Dr. C.A. Swahsoh and E. Gerlach f 6 r t h e i r c a r e f u l , reading of the : d r a f t : f 0 ^ o f t h i s t h e s i s . Furthermore, the f i n a n c i a l a s s i s t a n c e of the N a t i o n a l Research C o u n c i l through a s c h o l a r s h i p I s g r a t e -f u l l y acknowledged. 1. 1. I n t r o d u c t i o n The eigenvalue problem f o r the L a p l a e i a n ( - A ) , namely Au(x) + Xu(x) = 0 x € E u(x) = 0 x e 3E, has "been the subject of much d i s c u s s i o n i n the l i t e r a t u r e . Courant and H i i b e r t [ 5 ] e x t e n s i v e l y i n v e s t i g a t e d the problem when the domain E i s bounded. Tltchmarsh [10]> when he discussed the SchrKdinger equation gave an a l t e r n a t e treatment of the problem and extended many of the r e s u l t s t o the whole plane. R e i l i c h [ 9 ] was t h e f i r s t t o o b t a i n results"when the': domain E was unbo d i f f e r e h t from the whole plane. He ."showed t h a t when the doM^ along a given ray, ;• thetp^blpbem^vhadTarSi:B^ete.;::set of eigenvalues; Mol&anov [8] gener-a l i z e d R e l l i c h ' s i r e M l t t o i n c l u d e 'domains which were "narrow at i n f i n i t y " i n a wider sense, S p e c i f i c a l l y , those domains which do not co n t a i n ^ i n f i n i t e l y many d i s j o i n t s p h e r i c a l b a l l s jDf eq\aal " p o s i t i v e r a d i u s . <31£zman: i n h i s recent book [7]•" gave a d i s c u s s i o n of problems c l o s e l y domie>ted t o the eigenvalue problem f o r unbound-ed domains. S t i l l more r e c e n t l y , C l a r k [4] published a 'paper "An embedding theorem i n f u n c t i o n spaces" which allo w s : t h e treatment '"of*an: a r b i t r a r y e l l i p t i c , o p e r a t o r i n s t e a d of the L a p l a e i a n . The present d i s c u s s I 6 n ; w i l l center around the existence and /properties 'of the, Green's f u n c t i o n ( G P ) f o r the, problem. The, general method^?Of a t t a c k and the arguments I n many cases 2. f o l l o w those used by Titchmarsh [10]. The f i r s t r e s u l t , whose.proof i s an a p p l i c a t i o n of the A s e o l i - A r z e r a theorem t o the w e l l known case when the doiakiri I s bounded, i s the e x i s t e n c e of the GEV I t i s important to note t h a t although t h i s t c ^ h s t r u c t i p n gives many of the p r o p e r t i e s :1«^ch a GP must p'oisseisSj i t does not show the. GP tends to zero at the boundary. Prom this;:'construction a b a s i c r e s u l t e a s i l y f o l l o w s , namely th a t the GF I s bounded by the fundamental s i n g u l a r i t y f o r the domain. The!fundamehtal singularity ( a B e s s e l f u n c t l o h ) I s the GP f o r the rprdblem; on the. whole space; T h i s r e l a t i o n , which i s hot needed i n Titchmarsh's development, forms a c r u c i a l l i n k i n the present a r ^ I n P a r t 3 we d i s c u s s the eigenvalues and e i g e n f u n c t i o n s of th^;^prbblem,;br more p r e c i s e l y , those :of the operator T defined by o 8 ( T ) H ^ ( E ) n {f e ; L 2 ( E ) | Af e L ^ E ) } Tf « -Af when f e $ ( T ) , where , E ;' 'igat.M^ -e.s.' -tne.. "narrowness at I n f i n i t y " c o n d i t i o n given by C l a r k [ 4 ] , '„• The main r e s u l t t o be : proved In t h i s section I s that the eigehfunctldhs; of T are'} contained i n H^(E) . . Prom . t h i s r e s u l t we conclude t h a t 1 X b -» as- X -» « f o r each n, where the X.„ „ are the 'eigenvalues :for. tjbe'.truncat<ed .domain E v (see P a r t I I ) and the X are the eigenvalues f o r T defined A . ".- n on E. T h i s r e s u l t appears to be new but not unexpected. T h i s s e c t i o n cohcludes w i t h some d i s c u s s i o n about the e i g e n f u h c t i o n s near the boundary. At v a r i o u s ' p o i n t s i n the argument r e s t r i c t i o n s are a p p l i e d to the domain„ Most of these r e s t r i c t i o n s are Smoothness condition's f o r the boundary; however the;most important r e s t r l c t i o n y whlteh i s Introduced i n P i r t : 4 , i s an assumption about the r a t e at which the domain narrows;a T h i s c o n d i t i o n i s a's' f o l l o w s : f o r ; example, when dimension of E=2, l e t the boundary Tof E be the p o s i t i v e ;; x ^ - a ^ i s and the set {x^, ep(x^)} x 1 > 0 , where cp i s a p o s i t i v e :cT&htiniibu.'si'-furi^tlotf;'.•' then we assume t h a t there e x i s t s an i n t e g e r k; such t h a t qp^Cx )^ i s I n t e g r a b l e t o i n f i n i t y . I t does riot Seem p o s s i b l e toi; t r e a t the GP d i r e c t l y as .Tltchmarsh doeSi i n s t e a d i n P a r t 4 we consider the i t e r a t e s ; o f the GP, which are much smoother. An important theorem i n t h i s re^sfect: IS t h a t there;\e^Br%B"'rJ^: ^ %e^jB^^:\ot the GP which i s a H i l b e r t - S c h m i d t k e r n e l . Prom the H l l b e r t - S c h m i d t p r o p e r t y we o b t a i n t h a t the s e r i e s £ converges f o r some k which IS dependent on the narrowness of E. Once i t i s known tha t t h i s s e r i e s c o h ^ t h a t the i t e r a t e which i s H i l b e r t -S,chml,dt actually^ tends to Z B r d ; o n the boundary. T h i s r e s u l t ^ as we" show;iri P a r t 5 > i n t u r n i m p l i e s ; l i h a t t h e GF i t s e l f tends t o zero" at the boundary and 4. The. l a s t s e c t i o n , P a r t 6, d i s c u s s e s a p p l l c a t i o n of the GF t o asymptotic prohlems f o r the eigenvalues and eigenfunetidhs. The f i r s t a p p l i c a t i o n Is.',a proof of the asymptotic formula •Jtytt*) ~ ^ J T(X) dx " E (See Theorem 6.1 f o r n o t a t i o n ) , which reduces t o the w e l l known formula of Weyl, I.e. N( X) ~ TRF 1 a r e a E when the domain E has f i n i t e area. T h i s asymptotic formula was f i r s t g i v en hy C l a r k [3] f o r a smaller c l a s s of domains. The second a p p l l c a t i o n proves formula due t o C'arleman f o r .the eigenvalues extends t o unbounded E. Throughout the work a l l lemmas and theorems are numbered s u c c e s s i v e l y , f o r example Theorem 2 . 4 means Theorem 4 P a r t 2 . Within; a section-/ e q ^ t l d h s and d e f i n i t i o n s are r e f e r r e d t o by a number but i f w e wish t o r e f e r to eduation 6 of P a r t 4 , then we would w r i t e equation; ( 4 . 6 ) :. References are r e f e r r e d t o by the author's name anci, a nuiber c d r r e s ^ b M i n g t o h i s paper i n the b i b l i o g r a p h y at the end. P a r t 2. C o n s t r u c t i o n of the Green's f u n c t i o n and some of I t s elementary p r o p e r t i e s . I n t h i s s e c t i o n W e s h a l l g i v e a c o n s t r u c t i o n of the Green's f u n c t i o n f o r the L a p l a c l a n on domain which w i l l he defined below. Notations; L e t R n denote E u c l i d e a n n-space, l e t x=(x^, x 2 , ..., x^) denote a t y p i c a l p o i n t I n P^ and l e t |x-y| be the E u c l i d e a n norm. L e t E denote a simply connected unbounded domain R n, w i t h boundary 3E. For a given E d e f i n e E' = {p e E | d(p, 9E) > b j , where b I s a f i x e d p o s i t i v e constant which determines E', and d denotes d i s t a n c e . Furthermore, l e t L = {X e € X.| < K, and |lm X| > v or Re \ < -2v}, where K and v are f i x e d i p o s i t i v e constants which determine L. We w i l l sometimes w r i t e L=*L( v) t o I n d i c a t e the dependence of L on v. F i n a l l y we set E x = {x e E |x ±| < X, i = l , 2, ...n} X > 0 . Consider the boundary value problem A u(x) + X u(x) = 0 x e E (1) u(x) = 0 x e SE 6. where/ A denotes the L a p l a c l a n . We d e f i n e the operator T i n L 2 ( E ) by: <£>(T) = H 0 ( E ) n [f e L 2 ( E ) | Af e L g E ) } f (2) Tf = -Af i f f € o9(T) where H^(E) denotes the standard Sobolev space (see e.g. Dunford and Schwartz [6> P • 1652]) w i t h the norm ilfllm - [ J I | D a f ( x ) | 2 d x ] 1 / 2 (3) • E |a|<m i n which we use the standard n o t a t i o n s =1x7 i f K-^ 1 • B n a n a = (a-L, a 2 , .. a n) j> (4) I ct J - Z . The c l a s s i c a l theory f o r problem (1) s t a t e s t h a t i f E ' i s bounded, then T i s a s e l f - a d j o i n t operator i n the H i l b e r t space L;2( E ) , and has a d i s c r e t e spectrum boh s i s t i n g of i s o l a t e d , e^ig^hyalues X n, n=l, 2, 3, ... e t c . , each eigenvalue c o r r e s ^ ponding t o a n e i g e n f u n c t i o n u - ( x ) . L e t us assume tha t E x : I s s u f f i c i e n t l y r e g u l a r t o a l l o w us t o construct a Green's f u n c t i o n G x(x,y,X) f o r problem (1) i n the sense of Titchmarsh [10, Ch 14]. I n the present s e c t i o n We s h a l l need the f o l l o w i n g ; properties'/."of G x(x,y,X) s 7 . G x(x,y,X) has a standard s i n g u l a r i t y f o r x=y. (See Lemma 2 . 7 ) . I f xie E x> X ? some L, then G x(x,y,X) tends t o zero as y approaches the houndary of E x . I f we def i n e the operator G x ^ by G x x f (x) = J G x(x,y,X) f ( y ) dy f ve L 2 ( E X ) (5) E X then we need two theorems: G X X• "n-W "''BS (^n - ^ ) " ^ u n ( x ) f o r X non r e a l (6) and l iO x,. xf I! v " 1 ||f I! where X € L( v) . (?) A l l these r e s u l t s are proved i n TItchmarsh [10, Ch 14] except f o r ( 7 ) , which i s our f i r s t lemma. Lemma 2.1. Suppose th a t e i t h e r |lm x | >_ v > 0 or th a t Re X < - 2v then llGx^fll"•'.< v " 1 llfll . (8) Proof. . We'^ knowT-itrpra' t h e c l a s s i c a l theory t h a t G x ^ i s the re s o l v e n t operator of a non-negative s e l f - a d j o i n t operator T i n L 2 ( E X ) i . e . G x^ x = ( T - X I ) " 1 . Now i f X=a+i0, al<~-2v C O and |p( < v, then 8. Hi*-xx II > ||T+2vi|| - lleill > ||T+2vl|| - v . But since TX), T+2vI.\> 2 v l , so t h a t ||T+2vl|l> 2v. Hence IIT-XIH > v, and t h e r e f o r e ||Q X ; XII = IKT-XI)" 1!! < v" 1 , which i s equivalent t o ( 8 ) . One can e a s i l y show, Using the r e s u l t s \ o f Titchmarsh [ l O , Ch 12], t h a t (8) holds f o r |lm X| > v > 0, completing the lemma. The f o l l o w i n g c o n s t r u c t i o n of the Green's f u n c t i o n f o r the problem ( 1 ) w i l l he given only f o r the plane. A s i m i l a r c o n s t r u c t i o n i s a v a i l a b l e f o r higher dimensions. The argument, up t o a p o i n t , w i l l be s i m i l a r to'Titchmarsh's argument f o r the whole space R 2| however, the presence of the boundary of E requires;Us•to be much more p r e c i s e about the nature of the convergence. L e t s— 2 ( h u<* I - \ (i -15>i i f «•<•» g(x) « g(x,u) = < / 0 i f r > R, where r=|x-u|. We remove the s i n g u l a r i t y from G x(x,y,X) by s u b t r a c t i n g the f u n c t i o n g ( x ) ; we set T x(x,y,X) G x(x,y,X) - g(x,y) (9) Note: P Y i s dependent on R, since g i s . I 9 . Theorem 2.2. The set [X|X>0] has a subsequence ^ X k ^ Xk~* such t h a t the s e q u e n c e [ Q v ( x , y , l ) ] converges polntwise to a x k f u n c t i o n G(x,y,X) f o r a l l x, y e E and a l l X not on the non-negative p a r t of the r e a l a x i s ; Furthermore, given E' c E, L( v) and X > 0, the sequence; [ p ^ X y y , X) ] converges u n i f o r m l y f o r x, y e E x and X € L( v ) . Proof. FIGURE 1 L e t f ( x ) = G x(x,y,X) and 'g-'('x') he as above; f o r x, y, u e E x w r i t e r = |x-u| and r ' = |y-u|, as i s f i g u r e 1 L e t the c i r c l e |x-u| ==r<R be contained i n E x . By Green's formula f o r the re g i o n r^R ( c f . Titchmarsh [10, p. 3+]), we have (u,y,X) - g(u,y) = = - i : f G Y(x,y,X)dx + r<R + X J g(x,u) G^XjyyX^dx . (10) r<R 10. L e t P Y(x,y,X) = G Y(x,y,X) - g(x,y) f o r t h i s R, and F(x,u) = S u b s t i t u t i n g these r e l a t i o n s i n t o (10) ,>we o b t a i n Px ( u , y , X ) = J G x(x,y,X) F(x/u)dx E X = (Gx(.",y,x), P(-,U ) ) . The f i r s t problem i n the proof i s t o show tha t P x(u,y, X) i s uni f o r m l y bounded f o r u, y e EX and X e L ( v ) (where X_ o - , ° ° ahd v Q are a r b i t r a r y f i x e d c o n s t a n t s ) . We f i r s t apply the Schwarz i n e q u a l i t y t o P x t o o b t a i n ir x(u,y,X)l < l|Gx(.,y,X)H l l F(-,u)||. (11) For the Second norm i n (11), we have |gix,u; 1 2 r<R r<R lF('.;yuj||2 <:-4TT f <3X + 2|X|2 f ( x , u ) | 2 dx , ~ ir R J J from the d e f i n i t i o n of F ( x , u ) . Therefore I!F(SU ) | ! 2 < K(u,R ,|x|) , where K(u,R,|x|) i s bounded i f u and X are bounded, the di s t a n c e between the boundary of E and u i s gre a t e r than R, and R i s bounded away from zero; The f i r s t term i n (11), namely ||GX( • ,y,X) ||, i s more d i f f i c u l t to estimate. One proceeds as f o l l o w s : Observing t h a t the form of P(u,y,X) s a t i s f i e s 11. Lemma 2.1 (note P x(u,y,X) = G x ^F(u,y)), we have I! r x(u , . ,x)l! 2< v - 2 ||F(-,U)!|2 (12) < v" 2 K(u,R,|x|), where eithe r |lm X| > v > 0 or Re X <_ -2v and v > 0. From the d e f i n i t i o n ; o f P x i n (9) one has !JGX(^,-,X>(i2 < 2|1 r x(u,,X) ||'2' + 2||g(ui Oil2, from which i t follows that |l.Gx(u,-,X) ||2 < (l+v~ 2) K ( u , R , | x | ) (13) In these formulas i t Is important to note that K i s independent of X. Combining these with the Schwarz inequality above, one obtains the f i n a l estimate I r x(u,y,X) | < f(l±v" 2) K(y,R, | X|) K ( U , R , | X|) ] l / 2 , where y and u cannot be closer to the boundary of E than the distance R. We can shorten t h i s i n t o I r x ( u , y A ) | i • • , K ( i i y y > . R , ] x ' | ) ( 1 4 ) where K Is bounded i f f o r some given E', X^ and L . u, y e E o o and X e L Q . We now apply the same argument as Titchmarsh [10, p. 35]* t o a r r i v e at where |u'-u| < 6 = 6(e) independent of ( X ) I n order to:do this,one J u s t needs t o consider the r e p r e s e n t a t i o n of P x , equation (10.), and make appropriate estimates u s i n g the X-uniform hound (14). By 'symmetry a s i m i l a r r e s u l t holds f o r the y v a r i a b l e We now have the d e s i r e d e q u i c o n t i n u i t y I n u and y, hut we must also: have i t i n the X v a r i a b l e , i n order t o apply the A s c o l i - A r z e l a theorem., . T h i s I s achieved as f o l l o w s " by the r e s p l v e n t equation G x(x,y,X) we have D x r x 8 3 fix ®x = (Gx(u*'->'^>"• %(• *y A ) ) • The r i g h t hand :sid£ I s hounded, as" X tends t o I n f i n i t y , f o r u , y € E^. and X e L .,• by the* estimate (13) and the Schwarz X o ° ; i n e q u a l i t y . Since t h e p a r t i a l d e r i v a t i v e ; of /"^ w i t h respect to X i s u n i f o r m l y bounded, P x w i l l be u n i f o r m l y continuous I n X. The above c a l c u l a t i o h s and remarks show t h a t the set of f u n c t i o n s [ P x ( x , y , X ) 3 as X — *, f o r x, y e E£ and X e L Q " o ( E 1 , X Q and L Q being f i x e d ) , i s equicontinuous I n each of the three v a r i a b l e s s e p a r a t e l y . The A s c o l i - A r z e l a theorem says t h a t such a set i s compact, i . e . there e x i s t s a f u n c t i o n P(x,y,X) such t h a t P v (x,y,\) tends t o P(x>y,X), uniformly f o r some A k subsequence {X^} tending to I n f i n i t y , when( x, y e Ejr and X € L . ;• O Now by a simple d i agonal Iz at i o n process, we can o b t a i n a 13. sequence [ p Y • f*',y,"X) ] which converges (polntwise) f o r a l l . . . . . A k x, y e E, and V e"C w i t h X not O H the non-negative r e a l a x i s . Namely, l e t X . < X , < . . . - » » o 1 E l c E| c ... E o 1 L Q . c L 1 c ... - {X | Im x4=© or X < 6 ] . -1. We could choose f o r example; Ej to he E' = {peE d(p , a E)>n } etc . L e t [X(G , n )] be a sequence approaching I n f i n i t y such that [ Px'('6'-n)'^ converges u n i f o r m l y when x, y and X are r e s t r i c t e d to E^, XQ- and L.^. L e t [ X ( l , n ) ] be a subsequence of [X(6,n)i such t h a t f nx(i n ) 3 converges u n i f o r m l y f o r . x,y> X r e s t r i c t e d to E£, X^ and L^, and so on. Then the diagonal sequence X* = [X(n,n) ] I s such t h a t [ P x *;] converges;polntwise f o r d e s i r e d values of x, y, X. We note t h a t t h i s sequence a l s o has the property t h a t , given any E», X and L ( v ) ^ [ P x l converges un i f o r m l y f o r • x, y e ,E X and X: e L( v ) . T h i s completes the proof of Theorem 2.2. A cohseqiiehee of the above theorem I s t h a t we have a Green's f u n c t i o n f o r E w i t h the r e p r e s e n t a t i o n ©(*,y>X) = P(x,y,X) - g(x,y) . i^rth e r m b r e , p s a t i s f i e s the I n t e g r a l equation r(v,y,\) = J G(x,y,X)dx + X J g(x,u) G(x,y,X)dx ( l 6 ) • l f R r<R r<R 14. where r=|x-u| and R i s so small that the c i r c l e |x~u|<R i s contained i n E. We s h a l l now examine some of the elementary p r o p e r t i e s of the Green's f u n c t i o n . Theorem 2.3. G(x sy,X) i s continuous f o r x=f=y and i t i s such t h a t G(x,y,X) - l o g ~ + 0(1) as p - 0 where p = |'x-y|, and a l s o ..a. G(x , y A ) = _ J L . + 0 ( i ) as p - 0. dp 27rp Theorem 2.4. G(x,y,X) has continuous p a r t i a l d e r i v a t i v e s up t o the second order except at x=y, and {A + X] G(x,y,X) = 0 i f x + y • The above two theorems are proved i n the same manner as i n the case of bounded E> Theorem; 2.5. | l G(u;.,X)|! 2 :< v " 2 K(u,R y|X|) where K i s bounded i f u e E' and X e L r t(v) . ° Proof. By i n e q u a l i t y (11) we have lir x(u, •, X) (I2 .< v" 2 K(u,R, | X|). Then f o r EV and X Q f i x e d (E*=E' X) o j I T x(u>y,X)| 2dy < v" 2 K(u,R,|x|) (X>X0). E* F i r s t l e t X -• » through the sequence defined i n Theorem 2.2. We then have J I r(u,y,X)| 2dy < v ~ 2 K(u,R, | x|) E* where the r i g h t hand s i d e i s independent of E*. Thus, l e t t i n g E* tend t o E, one has hy Fatou's lemma || P(u,-,X)|| 2 < v " 2 K(u,R,|x|) . I f we combine t h i s I n e q u a l i t y w i t h the d e f i n i t i o n of P and the r e s u l t ||g(x,-)ll 2 = AR 2, then Theorem 2.5 w i l l f o l l o w ; We d e f i n e the operator G^ as f o l l o w s G x f ( x ) = J- G(x,y,X) f ( y ) dy f e L 2 ( E ) . E The I n t e g r a l e x i s t s i n view of equation (13) tor a l l xeE and a l l X contained i n some L. Theorem 2.6. : I f H(x,X,f) = - G x f ( x ) , where f € L 2 ( E ) and X I s not on the non-negative r e a l a x i s , then ||H(.,X,f)|| < v' 1 llfll , U+X] H(x,X,f) = f ( x ) and H(x,X) = 0( | X | 1 / / 2 v _ 1 ) un i f o r m l y f o r x e E Y , |X| > 6 > 0 and Im X = v =(= 0. We s h a l l prove the f i r s t r e s u l t and remark t h a t the other two are proved by the u s u a l methods. (See Titchmarsh [10, Ch 12]) . Proof. The f i r s t item to show i s t h a t H" x(x,X,f) - H(x,X,f) 16. u n i f o r m l y x € E£ , where H" x(x,X,f) = -G x ^ f ( x ) . low o f Hx-H = J G(x,y,X) f ( y ) dy - J G x(x,y yX) f ( y ) dy . E Ex I f we set Gx=0 outside E x we have %-H - "J [G(x,y,X) - G x ( x 5 y , X ) ] f ( y ) dy + J [G(x,y,X) - G x(x,y,X)I f ( y ) dy; E-E* E* f o r any E* equal t o some E i . By the Schwarz i n e q u a l i t y the second term tends t o zero as X -» » since G x tends t o G u'tilfbrmly over E*. Applying the Schwarz I n e q u a i l t y t o the f l r s t term, one has | f i r s t term) J |G(x,y,X) .- Gx(x,y,X) 12dy • J | f | 2 d x E-E* E-E* * {2| | G(x,.:,x ) l i 2 : + 2||GX(X^,X)!I2} J | f i 2 d x . E-E* T h i s term then can he made small '''uniformly i n X hy n o t i n g f e L 2 ( E ) , x e E' and the r e s u l t s of equation (13) and Theorem 2.5". We how have the de s i r e d convergence f o r . H x. I f X e L(v) and E* c E x , we have by Lemma 2.1 J * |H x(x,X , f)T 2dx 1 v ~ 2 J | f | E* EJ' 2 d x <: v- 2 J | f | 2 d x = v " 2 ||f|| 2 . I f ; we l e t X '-» » through the sequence defined I n Theorem 2.2, IT* we have by the uniform convergence f o r H^v proved above, J :|H(x,X,f)| 2dx .<: v " 2 | | f j j 2 . E* I f we l e t E* tend t o E, we have by PatoU' s lemma i|H(-;x,f)i| < v" 1 ||f!|. Q.E.B„ Remark; At t h i s ' p b i h t irt the argument we do not know the boundary behaviour of G( x i y , X ) . Although we know that- f o r each f i x e d X, G^(x,y,X) goes to zero on the boundary as y tends t o the boundary, our convergence theorem (2.2) i s not strong enough to imply the r e s u l t f o r G(x,y, X). We now t u r n our a t t e n t i o n t o t h e fundamehtal s i n g u l a r i t y f o r the domain and i t s r e l a t i o n t o the Green's f u n c t i o n . Here we consider the case of a gteneral dimension n. Lemma 2.T. ;(Brownell [2, p. 555, Lemma 2.1]) ; There e x i s t s a r e a l p o s i t i v e f u n c t i o n H n m(r) defined f o r ; a l l r e a l , p o s i t i v e r and a), and a l l i n t e g e r s n>l, which i s " r e a l - a n a l y t i c v l h ty and stibh t h a t the f o l l o w i n g holds?; 0 < H ( r ) < M„ r~^ n" 2^ exp(-wr/ 4 ) n -> 3 0 < HI:. ( r ) < M p 1 + lo g ( l + ( u ) r ) - 1 ) e x p ( — 1 8 . where the, M n's are constants independent of u> and r , and o*n i s the volume• o f - t h e U n i t h a l l i n R.. The H ( r ) so defined n n, ar ' i s c a l l e d the fundamental s i n g u l a r i t y f o r A-iu . /For t h e / d e t a i l s of the proof see Brownell's paper [2, p. 555]. •'r,ThWoiejd:^B:. Suppose th a t G(x,y,X) i s obtained as a poln t w i s e alml.t"df f u n c t i o n s G x(x,y,X), as " i n Theorem 2.2 :f or n=2. Then 0 < G(x,y,X). :<; H ^ / p ) < K(p : «) where p=|x^-y|, K(p<») I s the bound f o r . f p ) given i n 2 Lemma 2.7, and X=-tt) where cu > 0. Proof. Since E x i s bounded the makimum p r i n c i p l e can be a p p l i e d t o prove 0 < G x(x,y,\) < H n ^ ( p ) , but f o r f i x e d x, y e E, G x(x,y,X) tends to G(x>y,X) as X goes through the sequence defined i n Theorem 2.2. Thus 0 < G(x,y,X) < H .(p) a l l x, y e E. Note t h a t t h i s r e l a t i o n i m p l i e s t h a t G(xyy,X) tends t o zero, f o r f i x e d x and X, as y tends t o i n f i n i t y . Theorem 2.9. G(x,y, X) = G(y,x,X) x, y e E. 1 9 . Proof. We know that G x(x,y,X) = G x(y,x,X) f o r x> y e E and X ; s u f f i c i e n t l y large. Since the l e f t hand side tends t o . ®(x,yyX) and the ri g h t hand / side tenCs to G(y,x,X), as/ X f tends 1 to i n f i n i t y V the: r e s u l t follows. The; f i n a l r e s u l t i n t h i s Section i s Theorem 2 . 1 0 . G(X y y>X) = G^2^ (x,y, X) and G(x>yy X) == hi G^n+^") (x>y/ X) fo r X a negative number and x, y e E. Proof. F i r s t we must e s t a b l i s h the "resolvent equation"? .; (X-X») (G(-,x,X), G( •yyA') |= G(y,x,X) - G(x,y,X'-) ' ( 1 6 ) f o r X and X' negative. The usual ; proof of ( 1 6 ) requires that G(xyyyX) be zero oh the boundary; however we want this;lemma independeht of the boundary r e s u l t so we proceed d i f f e r e n t l y . We have (X-X») J G x(s,x,X) G x(S,y,X')ds= G x(y,xyX) - G x(x;y,X') ( 1 7 ) by Green's theorem, since E x i s bounded. Let x, y € E*; where E*: IS some fixed E{ c E. The ri g h t hand side of ( 1 7 ) converges ': • • xo • 2 0 . to the r i g h t hand side of ( 1 6 ) by Theorem 2 . 2 . Thus we need to show tha t the d i f f e r e n c e (18) tends t o zero as X «. (Here we extend G x(s,y,X) by zero t o s £ E x . ) ; (G('xVi),. G(- ,y, X 0 ) - (G x ( ' >*i X.)' * G x ( • >Y> X')'.) ..-i ( G p,x,X) - G x(.,x,X), G(.,y,X«)) + ( l 8 ) • + (G x ( - ,x,X), G(-,y,X') - G x(.,y,X')) • By the Schwarz i n e q u a l i t y the f i r s t term of t h i s expression i s l e s s than l!G( -,x,x) - G x(>,x/X) j] 2 • ||G(*,yX') II2 . ( 1 9 ) The second f a c t o r of ( 1 9 ) i s bounded since y e E* and X' i s f i x e d . ITbw cohsideir the f i r s t f a c t o r of ( 1 9 ) / which equals J J G(s,x,X) - G x(s,x,X) I 2 d s + J |G(s,x,X) - G x(s,x,X) | 2 d s . ( 2 0 ) E-E* E* The l a t t e r term of ( 2 0 ) goes t o zero f o r each f i x e d E* as X goes'; t o I n f i n i t y , "since" G x converges u n i f o r m l y bh E*. Time r e s u l t w i l l now be complete I f we can show tha t the f i r s t term of ( 2 0 ) can be made s m a l l , independently of X, by an appropriate choice of E*. | l s t term of ( 2 0 ) | < 2 J | G( s>x, X) | 2 d s +. 2 J | G x ( s,x, X) | 2 d s E-E* E-E* < k J |K(pU))| 2ds E-E* by Theorem 2 . 8 , where p = | s-x| and X = -to , to > 0 . Thus the 21. f i r s t term of (20) can "be made small by p i c k i n g E*, arid the choice of E* w i l l be independent of X. A s i m i l a r argument can be stppii^d t o the second term i n (l8) ^  so tha t (18) approaches zero as X -» and equation (16) i s proved. From equation (16) we hsLve (G(-,x,X), G(-,y,X')) = ix-te)"1^^ ;-?^ ) - G(x,y,X'))-iTtiw$:,le%/ ^-)if the theorem i s proved. Note th a t G i s symmetric i n x and : y by Theorem 2.9. S i m i l a r p r o o f s w i l l show the r e s u l t s f o r higher i t e r a t e s . 22; P a r t 3. The EigenvaiUe.s of the Problem. | I n t h i s s e c t i o n w e S h a l l introduce c o n d i t i o n s on E t h a t w i l l alldW c p h s t r u c t i o n of eigenvalues and eigenfuhetions forjptdbiem: (2.1) I t t u r n s out t h a t a c e r t a i n c o n d i t i o n on E c a l l e d "narrowness at i n f i n i t y " w i l l be s u f f i c i e n t , provided E s a t i s f i e S x ' c e H a ^ R e l l i c h [9] and Mpieanov ;[8] gave "narrownessfat I n f i n i t y " c o n d i t i o n s s u f f i c i e n t n ^.oir':p:rbhl'em;'';'('2)l) to have a d i s c r e t e ; s p e c t r a C l a r k [4 ] gave a c o n d i t i o n which; we•Shall use t o construct the e i g e n f u n c t i o n and eigenvalues. The c o n d i t i o n ( c a l l e d c o n d i t i o n I ) i s as f o l l o w s ; I Corresponding t o each X>0 t h e r e e x i s t p o s i t i v e numbers d(X) and 6(X) s a t i s f y i n g a) d ( x ) ; + 6 (x) — b a s x — > b) d(X)'_••'/6(X) <M < » f o r a l l X c) f o r each x e E-E x there e x i s t s a p o i n t y such th a t |x-y| < d(X) and E n [z |z-y| < 6(X)'•} = 0. C o n d i t i o n I i n i p l i e s t h a t E I s narrow at i n f i n i t y i n the f o l l o w i n g sense; The set E i s s a i d t o be "narrow at i n f i n i t y " i f l i m p(E-E Y) = 0 , X-*» where p(A), f o r A: an a r b i t r a r y set i n R n, I s defined by p(A) = sup d(x>SA). xeA 2J. I t i s c l e a r t h a t p(A) i s the ;supremum of the r a d i i of the spheres in^ A. When T h ^ the operator T (equation .2.2), w i t h c o n d i t i o n I on E,'• we can concludes Lemma 3 . 1 . T I s a s e l f - a d j o i n t bperator I n the H i l b e r t space L 2 ( E ) ; the spectrum d(T) I s d i s c r e t e arid has no f i n i t e l i m i t p o i n t s ; f o r X £ CT(T) the re s o l v e n t bperator R^(T) = ( X I - T ) " 1 I s cpmpTetely continuous. /Remark; T h i s r e s u l t g e n e r a l i z e s the r e s u l t of R e l l l e h [ 9 , p. 335] t o &j.larger e l a s s ;pf dbmairis. L e t a(T) => [ X ^ ] , S}ffieTeSX£;<..\fe.,£ X^ ... e t c . Let \yr n and' n ( x ) he the eigenvalues and eigenf u n c t i o n s f o r the ' problem; •, j A u(x) + X u(x) = 0 x € E x u(x) = 0 x e a E x which are known to' e x i s t since E x , i s bounded. Since E x c E we have, by elementary v a r i a t i o n a l p r i n c i p l e s . ( c f e.g. Glazman [ 7 ] ) , X v •> X• . r I n view of the f a c t t h a t X v „ i s a hon-increasing X,n — n . • >• x,n • f U n b t l b h - o f X /: : f o r each f i x e d " n, we have •\x h "* *n••— x n as X - « f o r each n. (2) Lemma 3-2. ( T i t c h ^ a r s n [10, p. 334] theorem 22 . 1 4 ) . For p=l, 2y 3 , . ...... l e t H (x) have c o n t i n u o u s . p a r t i a l -24. d e r i v a t i v e s up t o the second order and s a t i s f y the d i f f e r e n t i a l equation (A + Xp - qjHp - t, where f and q have continuous p a r t i a l d e r i v a t i v e s of the f i r s t order. As p -» », l e t H p tend t o a l i m i t H(x) u n i f o r m l y over some given r e g i o n , and l e t Xp tend t o a l i m i t X. Then H(x) has continuous p a r t i a l d e r i v a t i v e s up t o the second order, and the equation (A + X - q) H(x) = f ( x ) i s s a t i s f i e d . The same r e s u l t holds i f we are merely given t h a t H i n mean square. Lemma 5 - 3 . There e x i s t s a set of f u n c t i o n s ( u n ( x ) } , such t h a t u v (x) tends to' u„(x) i n L 0 ( E ) f o r each n and some sub-x,n ' n x ' dy sequence of {X} tending to i n f i n i t y ( t h i s subsequence can be picked from the subsequence given i n Theorem 2.2); moreover A u n ( x ) + X n u n ( x ) = 0 x e E; u n has continuous p a r t i a l d e r i v a t i v e s up t o the second order; the u n are orthonormal, and X n are p o s i t i v e . Proof. Extend u x n ( x ) by zero the E-E x > Since 25-we have r2 , K , J l < 1 + h,n > and i f we take X > X . we have o Np#|fby;t^ 3 of C l a r k [ 4 ] , the embedding map H^(E) c Lg(E) i s completely continuous. Since by (3) the sequence u ^ ^ i s bounded I n H^(E), i t must t h e r e f o r e have a subsequehce convergent I n L 2 ( E ) . L e t u n ( x ) be t h e ^ i i m i t of t h i s convergent su^seqiienc^ 1; We s h a l l remove, by a d i a g b h ^ i i z a t i o n procedure, the r e s t r i c t i o n t h a t the sequence chosen may depend on n. For example, f i n d a subsequence X^ such t h a t [ u x ^\ converges',''-.then f i n d a sub-sequence Xg bf X^ such t h a t [ u x 2^ converges, and so on. ••• /• 2. .. We then take the diagonal of t h i s process.to show the r e s u l t ||u n(x)' -'.''u'x'^('x)f ••-»•;6". as X • -» » f o r each n. ( 4 ) Since we a l s o have X V„^X,. f o r each h, we can apply Lemma 3-2 A , H n t o o b t a i n c A u n ( x ) + X n u n ( x ) = 0 , x e E. Ljemiaa 3;2,alS6 s ^ s t h a t u R ; \ d ! i l have continuous p a r t i a l d e r i v a t i v e s ^ t o the^second order. Furthermore u n are orthonormal since < V um> = \ ± m (uX,n> uX,m> = 6m,n • A - * 0 0 2 6 . Since J I v u X , n l 2 d x = X X,n < h -n i f X ^ Xd> E X and D. u v ... -» D.u I n s i d e E, Fatbu's Lemma shows 1 x,n i n I f we now r e c a l l that X x n - X^ we can sharpen the r e s u l t (5) to ||7 u||2 £ X n by a simple c o n t r a d i c t i o n argument. From t h i s i t f o l l o w s t h a t a l l the X n . "are'non-negative;. Theorem 3.4; L e t E / be such t h a t through every p o i n t o f the boundary- passes a c i r c l e ; which l i e s otherwise e n t i r e l y i n s i d e E ,(roughly; t h i s Means t h a t the"point i s not the vertex of an outward-pointing a n g l e ) . A l s o , l e t E be "star-shaped". Then the " e i g e n f u n c t i o n " u : i s contained"in' H^(E); . n o' Remarks; 1. The proof w i l l be given f o r dim E=2, but holds f o r a l l dimensions; 2. The hypbthesis^ E, i s " s t a r shaped" can be dispensed w i t h e n t i r e l y by u s i n g a partition of u n i t y . 3. The hypothesis; t h a t the boundary of E has I n t e r i o r c i r c l e s , ; e t c . , can be weakened. The pro o f s of the above;three remarks w i l l not be given since Theorem 3.4 w i l l b e ; S u f f i c i e n t f o r b u r purpose as i t i s proved; Proof of Theorem 3.4. Write u=u n f o r the present. We noteSthat 2 f . u e H (E) "by i n e q u a l i t y (5) • The proof t h a t Titchmarsh [ 4, p. 9 9 ] gives to show tha t u n ( x ) -• 0 as x -* 3E f o r the case of E hounded vjorfcs f o r our ease a l s o . Thus, t o complete the pro o f , we need'to show t h a t u can be approximated by C*(E) f u n c t i o n s i n the ||||^  norm. We s h a l l approximate u i n s e v e r a l steps. Let g(S) be a f u n c t i o n such t h a t g(B) € G*(R^); 0 1 g(B) < l j g(B) - 0 f o r 6••'> 1; and g(B) = 1 f o r B < 0 , then d e f i n e g R ( x ) as g R ( x ) = g( |x| -R). We want t o show tha t llu-gj^ujl^ can be made small f o r s u f f i c i e n t l y l a r g e R. Now |u-g R u|j 2 = J*|u-g R u| 2dx + J l v C u - g g U ) | 2 d x E E (6) u-g Ru| dx + >R The f i r s t i n t e g r a l i n ( 6 ) i s l e s s than 4 v(u-gRu)I dx. >R u dx> sin c e >R 0 < g R ( x ) <. 1. Thus t h i s i n t e g r a l can be made small f o r l a r g e R since- u '€; L 2 ( E ) . Consider now the second i n t e g r a l i n (.6) v(u-gRu) | 2 d x < 2 >R | VU| 2dx + 2 • J >R |x">R |v(g Ru)|^dx. The f i r s t i n t e g r a l here can be made small since |vu| e L 2 ( E ) . Consider the remaining i n t e g r a l J I v(gRu) "dx x >R R+l> |v(g Ru)| dx x >R < 2 g 2 |vu| 2dx + 2 u |vg R| dx. ( 7 ) >R R< x <R+1 28. ''Thy-first-'integral i n (7) can be made s m a l l , f o r s u f f i c i e n t l y l a r g e R> "since g R < 1 and |vuj € L 2 ( E ) . Prom the d e f i n i t i o n of g R ( x ) w e M y e ' max |vgt,(x)| - max |vg( |x| -R) | R < | X | < R + I TT : R < ; | X - | - < R + I -^ : (max v |g'(|x| - R ) | max |g'(p) | . R<jxj<R+l 0<B^1 Hence | v g R | i s ; l e ^ R. Thus the^ l a s t integrai;LihvC7) can be made a r b i t r a r i l y s m a l l , from which i t f o l l o w s t h a t llu-gpull^ can be made small f o r s u f f i c i e n t l y l a r g e R , Thus we^may; ass:umey without Idss ;of g e n e r a l i t y , t h a t u has bounded IsuppdrtV T h e n e x t step i s t o snow t h a t u can be apprbximated, i n the H 113_ ho;M, w i t h f u n c ^ i n E. WeSkrib^ o r |xj>R+l^ Define; u to be zero f o r a l l other values of x outside E. Since E Is " s t a r - s h a p e d ^ we have t h a t i f x e E, then (1- £ )x e E f o r 0 <£< 1. This can be achieved by t r a n s l a t i n g (0, 0) i n t o E i f necessary. L e t u^(x) t= u((1-C )x) . By uniform c o n t i n u i t y u ( ( l - £ )x) - u(x) as S-Q. i f |x| < R + 1. Hbweyery s i n c e u e we have, by a f u r t h e r a p p l i c a t i o n of the p r i n c i p l e ;bf u n i f orm' c o n t i n u i t y y % u l "* D i u a s ^ "* 0 i = l , 2, ..., n. This means t h a t 2 9 . Iji^-ul^ = J |u-u 1| 2dx+ J Ivtu-u-^^dx ER+1 ER+1 can be made s m a l l , f o r s u f f i c i e n t l y s m a ll € , and hence we can assume, without l o s s of g e n e r a l i t y , that u Is i n (^(E). To complete the proof we need t o show th a t u(e .C*(E)) can be approximated i n the norm of H^(E). To do t h i s , l e t J^- be the m o l l i f i e r f u n c t i o n ( c f . Agmon [ l , p.5]).. A standard r e s u l t ( c f . Agmon [ l ] Theorem 1.5) shows t h a t u(x) e C^(E) f o r s u f f i c i e n t l y s m a l l £ . A f u r t h e r r e s u l t ( c f . Agmon [ l ] Theorem 1.10) shows th a t J<£ u tends t o u, as € - 0, i n the norm of ' H^(E); since u has compact support i n E. This completes the proof t h a t u e H^(E). Lemma 3.5. (The Pars e v a l formula). I f f e L 2 ( E ) and G n = (f,'-u )., then l l f l l 2 = I l c n l 2 • n=o Lemma 3.5 can be proved by s l i g h t l y modifying the proof i n Titchmarsh [4, p.104]. Remark: I f f and g are i n Lg(E) we can show, by a p p l y i n g the P a r s e v a l formula t o f + g, tha t oo ( f , g ) = I a n b n > n=o where a R = ( f , u R ) and b R = (g , u R ) . 30. A s - a . summary of the' pre deeding.results we'may State the f o l l o w i n g theorem: Theorem 3 .6 . L e t the domain E, he "narrow at i n f i n i t y " and s a t i s f y c e r t a i n r e g u l a r i t y c o h d i t i o i i s , then the eigenfunctioris constructed i n Lemma 3*3 c o n s t i t u t e a complete set of orthonormal f u n c t i o n s i n H^( E) "''''satisryl"hg;;the'~%#atibn' A u n ( x ) + X n u n ( x ) =. 0 f o r x e E and the boundary c o n d i t i o u n = 0 on the boundary of E. . '] In; futui*fe;,~when we r e f e r ; t o t ^ eigen-f u n c t i o n s of (2.1) wemean the e i g e n f u n c t i % r t s x a n d e i g e r t y a i u b s t o f T constructed i n Lemma 3.1 and 3.3- Note: since the eigen-f u n c t i o n s are complete we have shown \^ = X' n ( c f . equation (2) ) . I n the next;, theb important i n v e r s i o n ;' ;prbperty of Theorem 3.7. I f X i s riot ;bh\ the h^ then = ( 8 ) Proof. We s h a l l prove (8) by l e t t i n g X go t o i n f i n i t y i n F i r s t the r i g h t hand side of; (9) converges t o the r i g h t hand side of (8) by the r e s u l t s giveh e a r l i e r i n t h i s p a r t . ; Nbw consider the d i f f e r e n c e 31. ( G(x, - ,X), u n ) - (G x(x;- , X ) , ux^n). (10) We can consider U X n G x(x,y,X) t o be zero outside E x and thus we are able t o use E as the domain of I n t e g r a t i o n i n the second Inner;products Expression (10) i s now equal t o ( ( G - G x ) ( x , . , X ) , u n ) + ( G x ( x , - , X h u n - u x ^ n ) . (11) By the Schwarz i n e q u a l i t y the second term of (11) i s bounded by ! l G x(x,-,X)|| l l u n - u ^ J . The f i r s t term i n t h e above i s bounded by Theorem 2.2 equation 2.13 (i ndependent of X ) . Since H u n " u x n" goes to zero as X goes t o i n f i n i t y , the second term of (11) goes t o zero as X gbe>Vto i n f i n i t y . I f we l e t E* = E£ where E' and X Q are o a r b i t r a r y , theh the f i r s t term of (11) can be w r i t t e n as J [ G x - G ] u n dy .+ J [ G x - G ] u R dy . ( 1 2 ) ; E-E* • E* '-" The second term i n (12) goes t o zero f o r f i x e d E* by Theorem 2.2. By the Schwarz i n e q u a l i t y the f i r s t term i n (12) i s /less:'; than ( J I V G ' J u n * * } 1 / 2 E-E* E-E* r p 0 -.1/2 < | | ( G x - G)(x,.,X ) ! l • [ J u * d y j E-E* Now since | | G x(x,-,X ) - G(x, • , * ) ! ! i s bounded independent of X (Theorem 2.2 and 2.5) and the remaining p i e c e J u 2 (y)dy can E-E* 32. be made as small as we please ( s i n c e u n e L 2 ( E ) ), we have i l m <%n u X , n ( x ) = G X un( x>> completing the proof of Theorem 3*7. Wel'snall f i n i s h t h i s ; p a r t w i t h two lemmas on the eigen-f u n c t i o n s . Lemma 3 . 8 . u(x) -» 0 as k ./-» • where u i s any eigenf u n c t i o n corresponding t o an eigenvalue X. Proof. u has the r e p r e s e n t a t i o n ( c f . Titchmarsh equation 22.9.3) u(y) W - i ^ J u(x)dx + X J g(x,y) u(x)dx (13) i r R r<R ;5 r £ R 2 where g(x,y) = ^ [ l o g § - | (1 - ^ ) ] r < R = 0 r > R, arid r = |x-y|. I f we apply the Schwarz i n e q u a l i t y t o (13) s then l«(y)| < [ J* dx • J* u 2 d x ] 1 / 2 + X[ J u 2dx • J g 2 ( x , y ) d x ] 1 ^ r<R r<R r<R r<R - ~ (14) < [ T T " 1 / 2 R"1 + XAR] [ J u 2dx] r<R since J g 2(x,y)d;x = ( A R ) 2 where A i s a constant. R„ 3 5 . Consider the f o l l o w i n g diagram, where r <_ R c E ; FIGURE 2 Since uCy-^y-j) = 0 , we have u(y i x, y 2 ) * - J D t u ( y i , t ) d t . y 2 -Therefore y 3 u ( y x , y 2 ) | 2 .< ( y 3 - y 2 ) J [D t u ( y i , t) ] 2 d t y 3 < R J [D t u ] 2 d t . 0 H Q W , i n t e g r a t i n g t h i s i n e q u a l i t y w i t h respect t o y x from a t o b y, b, we get b J | u ( y i , y g ) ! ^ ! < R J J [ D t u ( y 1 , t ) ] 2 d t d y x a 0 I f we now use the f a c t t h a t |V u| € L«(E), we have 34. b | N y r y 2 ) I 2dy x < R o(l) as a - • . ' a If this expression ''idth:'re'^ e"'b't' to , y 2, one has r b j J |u( y i, y g ) ! ^ : dy 2 < R o(l)(d-c) < R 2 o ( l ) . . c a Thus, making the sides of :.'''khe'^¥qttare' touch the ci r c l e r <_ R (see figure 2), we have J |u(x) 12dx ..< R 2 o(l) as y -r<R Combining' this result With (14), we get |u(y)| < [ i r ~ 1 / 2 R _ 1 + X A R] [R 2 o ( l ) ] 1 / 2 as y -Hence i t follows that u(x) * 0 as x -» «. For the purpose of the next proof we assume (x,y) i s a point i n Rg. Lemma 3.9. Suppose that the,boundary of E i s givenflobalLly by y = F(x). Then for (x,y) i n a region which lEtclttdes a ;piece of the boundary [ x ; F ( x ) ] , |un(x>y) | <K ' ,X^,, where K i s Indepen-dent of (x,y) and n, arid dim E = 2. Remarks The prbof of t h i s lenMa i s / a remark [ 1 0 , p. 1 0 8 ] Proof; Cbhsider the foiibwlhg diagram 35-FIGURE 3. F(x)+36 - - - F(xV+26 F(x)+6 Ff where 6 i s s u f f i c i e n t l y s m a l l , i . e . we are c o n s i d e r i n g a small piece of the lower boundary of E. Since u x n i s zero on the boundary, we have y uX,n< x>y> 2 - f"J D t u X , n ( x ^ d t Thus u X , n ( x ^ ^ < 5 6 I [ D t u X , n ( X - ^ ] 2 d t by the Schwarz i n e q u a l i t y . I f we i n t e g r a t e t h i s expression w i t h respect t o y from F(x)+6 t o F(x)+28 , then F(x)+26 F(x)+25 y J ^ X j n ( x , y ) 2 d y < 35 ,1 dy J [ D t u x ^ n ( x , t ) ] 2 d t . (15) P(x)+6 F(x)+6 y1 I f we now interchange the order of i n t e g r a t i o n i n the l a s t i n t e g r a l , we see tha t y v a r i e s over a range not exceeding 36> thus the l a s t i n t e g r a l i s l e s s than y 9 6 2 J' t D t uX,n( x>*> J 2 d t -36. Next I n t e g r a t e ( 1 5 ) w i t h respect to x over [ x ^ j X j j t h i s g i ves x 2 P(x)+26 x 2 y J dx J' [ u x > n ] 2 d y '..< 9 6 2 J J [ D t u X j n ( x , t ) ] 2 d t dx. (16) x x P(x)+6 x r y x L e t us f i r s t consider the l a s t term of (16). We know from the case of hb^ded E x t h a t J I 7 ^ J * * * = ^ vX,n * E X Thus i f X > X-, then : —. O Hence l i m sup , f I v u v . I 2 dx •••"'.<. \ v : _ , Y _ . « J A i » n — A « » n X - » -E X hut since X Q i s a r b i t r a r y l i m sup j |v u Y _ ] 2 d x •"..<'. X„, Y - eo ± A > n ~ n E X since X Y „ - Xrt as X -».•>.. Thus the r i g h t hand side of (16) I s bounded by 9 6 X n as X tends t o I n f i n i t y , but since U v .-» u '•'• i h the mean, we have by Fatou' s theorem X,n n 37. x 2 F(x)+26 J - ; dx^ J [ > i i n] 2dy :< 9 6 2 ^ . - (17) x1 P(x)+6 We are now i n a p o s i t i o n to make the required estimate on u n . By/ T i t c h i r i a r ^ 22.$ .3] we have 7 r R r<R r<R K 1 where r </R :.; i s a c i r c l e [s ; | s-u| R] which i s contained i n E. Npw s u p p o s e that for x i n [ x ^ X g ] we have |P(x+h)-P(x) | J.2. vC'MIh| . Thus i f |x-x o| < £, then which Is; Irai turn l e s s than P(x^) + 56/4 i f Q £^6/k< /Likewise F(x) + 26 >; P(x Q) - MC + 26 > F(x^) + ^ 6. Hence the c i r c l e , center ( x Q , P( X q ) + -|6) and radius •^ 6 min (M"*1, 1), l i e s 'between the curves y = P(x) + 6 and y « P(x) + 26. \ Now, In expression (l8) set u = ( x & , F ( x Q ) + ^ 6) and R = ^ 6 min ( M - 1 , 1) X ^ 2 X ^ 2 . Note that the c i r c l e r < R w i l l be In the region of Interest since x j ^ 2 X^ 1^ 2 <.1 f o r a l l n. By the Schwarz inequality the f i r s t term i n (l8) i s . l e s s than - r ^ [ J dx • I u 2 ( s ) d s J 1 / 2  i r R rj<R r<R:;' 3 8 . which i n turn less than hy inequality (17)• Thus the f i r s t term of (l8) i s less than Mg, X^, Mere .Mg : iBlndependeht of 5, u and n. Again hy "the Schwarz inequality the second term of ( l 8 ) ;l;s?lefs::"thah. r<R R r<R where Cy are Constants independent of u, n, and 6. Gomhining a l l these results, we have J'u (x) | <M:2 X^  + C, X_ 6 . I f we now assume that 6 < 1, we have the desired i n — ' result l u n( x>y) I < K X n where K i s independent of (x,y) and 3 9 . P a r t 4 . The i t e r a t e s of the Green•s f u n c t i o n and t h e i r p r o p e r t i e s . For the sake of s i m p l i c i t y we assume i n t h i s p a r t t h a t the domain E has a boundary formed by the x ^ - a x i s and a f u n c t i o n , where cp(0) = 0, cp(x^) > 0 i f x^ > 0 , and ep(x1) s a t i s f i e s the smoothness c o n d i t i o n s i n Theorem 4 . 2 . We a l s o assume a l l c b n d i t l b h s i n p a r t 2 and 3 , so t h a t we s h a l l have a Green's f u n c t i o n G ( x : , y > X ) f o r E, eigenvalues X n f o r T> and eigen-f u n c t i o n s u n ( x ) . I f we are wbrking I n a v dimension n "_> 3 we s h a l l cohsider E t o be the domain I n R n formed;*by r e v o l v i n g ,cp(x^ ) about the x ^ - a x i s . B e f l n i t i l b h . L e t the i t e r a t e s i o f the Green's f u n c t i o n be defined as f o l l o w s ; rf1^ (x,y,X) = G(x,y,X) and ^ • c i ^ ^ i i l i ' A ) ^ # ( * i : S ' ; x ) , ; & ( - , ^ : ) > ( i i i ) . These i t e r a t e s are w e l l defined by Theorems "2 .5 arid 2 . 6 . Theorem 4 . 1 . The i t e r a t e G^ 2^(x,y,X) f o r dim E = 3 I s continuous on E x E and s a t i s f i e s 0<^ 2'C^ (1) where X - -m , w > 0 , and M I s a constant Independent of . x and y. 4o. Lemma. L e t E be an open s u b s e t o f a c y l i n d e r (of f i n i t e cross s e c t i o n ) i n R„. Then, as 6 •-» 0, ! • n • l i m J , ; , a f , ^ - ° ( X ) i f « + P < n u -»y E |x-u| |x -yr •> 0 ( l o g l / 6 ) i f a + B = ri = 0(6 n~ a"" B) i f a + B > n, where a + £ > 1 and 6 = |u-y|. Proof., I f u, y e E x , then the i n t e g r a l we are i n t e r e s t e d i n can be broken up as f o l l o w s : P dx + |* <3x / 2) J |x-u| a|x-y| P J |x-u| a|x-y| P EX+1 E~ EX+1 Since E x + 1 i s bounded, we can apply Titchmarsh [10, p. 3 2 3 ] t o the ' f i r s t i n t e g r a l i n ( 2 ) t o o b t a i n l i m f / ^ A = 0(1) a +.p < n U V y J |x-u| a|x-y|P ^X+l = 0 ( l o g l / 6 ) a + 0 = n = 0 ( 6 n _ a ~ P ) a + B > n . To complete the proof we need only show t h a t the l a s t i n t e g r a l i n (2) i s bounded. To see t h i s we note t h a t , f o r x e E - E x + 1 , |x-u| > 1 'arid - fx - y j ,>.1>. E ~ Ex+l i s contained i n a i tube of f i n i t e cross s e c t i o n , and a + 8 > 1 . Proof of Theorem 4 . 1 . By the property of the fundamental s i n g u l a r i t y f o r E (Theorem 2.8), we have 41. G ^ ( x , y , X ) <. -J K ( ( B | X - Z | ) K(u)|y-2|)dz, E where K(tup) = M p - 1 exp(-t«p/4) . I f we how apply the t r i a n g l e (2) i n e q u a l i t y t o the Integrand of ,G\. *, then a ( 2 )(xyy,X) < M2' W(- IIWI.) • J j x - z l f y - z l • exp(-1(1x^1 + |z-y|) ) E < M 2 exp(- « | W l / 8 ) J I x - z l f y - z l E < M» exp(- uu|x-y|/8) by the Lemma, since a + 0 = 2 < n = 3. Remarks; For dim E = 2 (proof below), we have G ^ ( x , y , X ) < M exp(- u u | x - y | / 2 ) i f m.y-1 . (3) However I f dim E = 4, and we t r y t o use the same method of proof as In dim E = 3, we get G ( 2 ) ( x , y v X ) < M exp(- «j|x-y|/8) l o g j-±^ry • CM Instead of (4) we can show G ^ ' ^ X y y / X ) < M exp(- t B | x - y | / 2 K ) (5) i f k > n, and i f dim E = n > 3- One proves t h i s fey a repeated a p p l i c a t i o n Of the method used t o prove ( 1 ) . Proof of (3) f o r dim E = 2 . Since, the Lerima does not apply t o t h i s case we proceed as f o l l o w s ; 42. We have 0 < G^ 2^(x,y , X ) < J K(t»|x-z| ) K(t»|y-z|) dz E ' where K(cor) ='f(tur) e - t u r , and f ( ? ) = M ( I + i o g ( i + r1)) / ( 1 + § 1 / 2 ) hy Lemma 2 . 7 . Consider the I n t e g r a t i o n above over three s e t s , namely N ] X , U ) " 1 6 > , N ( y , u ) " 1 6 ) and E*^= E^{H(x,tB~^6) U N ( y v a r 1 6 ) } , where N(x,-"a) = {z ': |x-z| < a}, and 6 < 1 i s such t h a t :|x-y| > 2 6 / We need the f o l l o w i n g estimates f o r f ( ? ) : I f ? > 6 , then f ( ? ) £ M ( l + l o g ( l + 6 " 1 ) ) « M 1, and i f § < 1 , then f(§) < Mg l o g How c^ ^ ( x y O B - ^ ) , which i s bounded by ' exp (-tejx - ^ I) J f(«j|x-z|) f(t«|y-z|) dz . N ( x , a j _ 1 6 ) I f z c N(x,to ~ " L 6 ) , then ti)|x^z| .;<•. 6 < 1 , from w h i c h i t f o l l o w s that f (u)| x-z| ),•<_ M 2 l o g | x - z | ~ 1, i f we assume «i > 1 . A l s o |y-z| >v26 - a>v*6 or uu| y-z | .o 5( 2 u u - l ) . .£ 6 , from which i t f o l l o w s t h a t f ( u u|y-z|) <. M 1« I f we s u b s t i t u t e these I n e q u a l i t i e s i n t o the i n t e g r a l oyer the set W(x,cu""16), then t h a t i n t e g r a l i s bounded by M,rM 2 f l o g dz , 43. which is.Ybounded/. independent ;o:f • oo and x, as long as uo >. 1, By symmetry a s i m i l a r r e s u l t holds f o r N(y,uu - 15) . Consider now'''the''.integral'over E*. I f z e E* then uo| x-z I > J and <»|y-z| > 6. Therefore the I n t e g r a l over E* i s l e s s than exp(-uu|x^y|/2)M2 | exp( ^ <u( | x-z|• +.,J-zVy|')/2) dz E* hy an argument s i m i l a r t o t h a t above. Hence the i n t e g r a l over E* i s bounded independent of uu, x and y i f OJ J>. 1, fi' <..l, and | x-yj j> 26. Combining these i n t e g r a l s , we have 0 < G.(2l(x,y,\) < M cexp(-tt)j;x-y|/2) i f o > 1 V (6) and |x-y| >. 26, where M 6 I s independent bf x, y arid ou. Since 6^ 2^(x>y,X) i s continuous at x<=y, the r e s u l t (3) i s proved. Theorem 4.2. i f E I s a plane r e g i o n such t h a t (a) there e x i s t s a t > 0 such t h a t o sup t > t — o < CD and cp(t) (b) f o r every B > 0, there e x i s t s a k^ > 0 such t h a t -pt cp(t) > kp e " p u t > t Q , st i n t e g e r s k .> 2, then there e x i s t s a constant or such t h a t f o r m> ou, and o — o 4 4 . I v | 02k>(x,y,X)| 2 dx dy < 2k, E E | l« p ( s ) | " \ S s : y 0 v where X = -to and I s - a constant d ^ e h d i n g only on co. Remark; The proof of t h i s theorem f o r . dim E 2 Is^due t o Clark. Proof. F i r s t l e t US; show-^e; theorem f o r k = 25 tfiai I s , ' c . b n M d ^ r G ^ ^ ( x , y - j i X ) . By a p p l y i n g the d e f i n i t i o n of t h e I t e r a t e s Of the ^  2.9 (symmetry of G(x,y,X) ), '•:dn'et can showV • E By Theorem 4 . 1 (Equation ( 3 ) ) We have Gi^(x>y,X).._.< M J exp(-co( |x-z| + |z-y| )/2)dz, ' E M O U R E ; 4 . : x=(x^:y5c2^ y = ( y ^ y 2 ) z ^ ( z p z 2 ) By c o n s i d e r i n g f i g u r e 4 and/applying the t r i a n g l e i n e q u a l i t y , we can see t h a t (x]y, X)' .••<; M j e " ^ 4 J J . exp(-(B| 'x^-z^ I ' A ) d z 2 d z 1 0 0 I f we c a r r y but the I n t e g r a t i o n w i t h respect t o z 2 we get » 5 . co-G ( 2 f )(x,y,X) <rH e _ a ) r / 4 J t p ( Z l ) e x p ( - u ) | x 1 - z 1 ! / 4 ) d z 1 . (?) 0 The remaining i n t e g r a l must be analyzed i n t w o p a r t s , the f i r s t p a r t being over [b, t ^ ] and the second :part over [ t Q , ,») 3 where t Q i s the p o s i t i v e constant given i n hypothesis ( a ) . L e t X. > t and considers v ' l — o t J ep(zt) e x p ( - ( f l | x 1 - z 1 | / 4 ) d z 1 t Q < M J * e x p ( - t o | x 1 - z 1 | / 4 ) d z 1 < VLX e _ a , x l ^ 0 X Mg OPCX )^ u s i n g hypothesis (b) . M, ffcjV and M 2 are: coristarits depending only on to. We now examine the second p a r t of the I n t e g r a l , namely 'the--'one over [ t Q , +«>) . F i r s t we note t h a t hypothesis (a) i m p l i e s t h a t f o r s u f f i c i e n t l y l a r g e K, -'(*)•;' max tp(t) exp(-K| t - ^ j ) = ^ ( t ^ for any tj_ > t Q . t > t„. —. o To see t h i s c onsider D t [ cp(t) ^ ( - E j t - t ^ ) ] := expC-Klt-^l)[V.q>"»--(t)--Kcp('t) ] < 0 i f t > t x and K i s s u f f i c i e n t l y l a r g e , and i f t Q < t 1 < t , where K i s s u f f i c i e n t l y l a r g e . ! 4 6 . T h i s proves (*). Appl y i n g (*) t o the I n t e g r a l Over [ t Q , > ) , we have: 00 J c p ( z 1 ) e i x p ( : - c f i j ^ ^O (» = J\'Mz±) e x p ( - 0 ) ^ - ^ 1 / 8 ) exp(-«)|z 1-x 1 |/8) dz^ < ipC^ j) J \"e^C-»r*i:-»iT/?)*i1 i M o t ^ ) tirhleh holds f o r Sufficiently l a r g e oo. Thus we have, combining the two estimates f o r the i n t e g r a l s , G ^ ( x , y , X ) < cp(x x) exp(-uor/4) (8) f o r x^ 2 t Q and s u f f i e i ent l y l a r g e "oo;. For reasons o f s i m p i i c i t y we Introduce the f o l l o w i n g functions r c p ( t o ) i f t <; t o •81) < q>(t) I f t;> t Q . Now by i n e q u a l i t y (7) we have .J G ( % x , y , \ ) £ M° exp(-»jr/4) i f ^ • .<; t Q . (9) Thus we have, by combining ( 8 ) and ( 9 ) , G ( 4 ' (x,y, X) X ^ ( x 1 j exp( -tor/4j, (10) ' 4 7 : where Consider how t h e ! of (10)"over E w i t h respect t o x. J | 4^(x,y,X) | 2 d x < M J | ^  ( x 1 ) | 2 exp(-«)r/2)dx E E • 0 6 M J"' 1 ^ 3 ^ ) | 3 e x p C - i B l x ^ j ^ l d x ^ 0 I f we apply the same a n a l y s i s t o ' t h i s as we d i d t o equation (7)> we have ^ [ ^ ( x y y A ) ] 2 ^ / I ^ l ^ ) ! 5 E f o r s u f f i c i e n t l y l a r g e w. I f we i n t e g r a t e " t h i s e g r e s s i o n again, t h i s time w i t h respect t o y, we have J J |G^>(x,y,X) | 2 dx-dy ^ J \Wy^\3& E E E 00 00 < M x J I ^ y i ) I ^  < Mg J- | cp(s) | \ s . 0 ' 0 Thus we have shown t h a t there e x i s t s a p d s i t i V e number such t h a t J* J | G C 4)c^y,X)| 2dx :dy 1, M. } | c p(s)| 4ds E E 0 f b r co > to... 48. I t i s now a simple matter t o extend the r e s u l t t o higher values of k. For example; G ( 6 ) ( x , y , \ ) = J G^)(x;z,X) G ( 2 )(z,yV\)dz . E By equations (3) and (10) we have 0 ^ 6 | ( x ; y i l ) < M expi-wr^k) exp^-iijr^/ajdz; E < M $ ( x x ) J exp(-«)(r 2+ri)/4)dz < M ; <p ( x ^ ;exp(^oor/8) , and t h e r e f o r e J J | G ( 6 ) (x;y ; , X ) I 2dx dy < M J J j $( X ] L) | 4exp(-i«r/4)dx ^dy E E E E OD < M J ; ! ^ ( s ) | 6 d s y 0 hy the usual arguments. The same c a l c u l a t i o n a p p l i e d t o l a r g e r values of ky arid the proof bf Theorem 4 .2 } i s complete, f o r dim E = 2. Remarks" bn extension .Of Theorem 4 . 2 t o dlmerislOris l a r g e r than two. The case dim E = 3;--I f we apply the same procedure to G^^(dlm E = 3) as we d i d t o G ^ (dim E := 2) u s i n g equation (1) i n s t e a d of equation ( 3 ) t we can show G( 4?(?yy,X) ^ M e~mr/Q J exp(-U J| x- L-z 11 / 8)dz . E 4 9 . I f we now r e c a l l t h a t i n three dimensions E i s the s o l i d - o f r e v o l u t l b h formed hy r o t a t i n g cpfx^) about the x ^ - a x l s , then by a change'•'of v a r i a b l e s i n the above equations and the usual . c a l c u l a t i o n s , we have <*> 2ir op(z^) G^ 4^(x,y,x) <M e' u , r^ 8 J* J\J e x p ( - a ) | x 1 - z 1 | / 8 ) r d r d > d z 1 0 0 0 i U ) _ / o " ! . cp( z_) < M e" r / ° J J exp(-u)|x 1-z 1 | / 8 ) d £ d z x 0 0 I f we I n t e g r a t e w i t h respect t o x, we have " j W(x,y , x)| 2dx < M J ' K ^ - V ^ E E eo 0 < M 2 ^ y ^ ) J |G ( 4)(x, x) | 2 d  $ ( X ] L ) 4 exp(-u)r / 4)dx E ' < M 1 J '.'$(x1)6 exp( ' .-co| x 1-y 11 / 4 ) dx^ 6 I f we i n t e g r a t e again, then , 00 8, l«PlSj i E E • 0 By s i m i l a r argument we can show t h a t J J I G(V (x,y, X) I 2 d x dy < J | q>( s) | 8 d s . J J | G ( 2 k l ( x , y , X ) d x dy < J | c p ( s ) | 4 k d s , E E 0 where t» > w\ f o r some tu and k > 2. S i m i l a r r e s u l t s hold — . o o — f o r dim E > 3. Henceforth we consider only domains E which s a t i s f y 50. the c o n c l u s i o n of Theorem 4.2. Thus there must e x i s t a constant k "such t h a t o I J I ^ V C x ^ - X ) ! ' 2 ^ dy < « 3 E E . ( k o ) that i s G must be a Hilbert-Schmidt K e r n e l f o r some k_. o Hote t h a t , although Theorem 4.2 i s stated only f o r domains which l i e above the x£-axis, l i t a l s o holds f o r domains'which have a s i m i l a r p i e c e below the x 1 - a x i s , f o r example, one described by -<p^ (Xj-') where cp^  s a t i s f i e s the same c o n d i t i o n s as cp. The next step i n our i n v e s t i g a t i o n " o f the boundary behaviour of G i s t o Show; t h a t G^ kl(x,y, X) tends t o zero a t the boundary f o r some k. In Order t o prove t h i s r e s u l t we s h a l l heed the p r o p e r t i e s of X^ and UJ^ (X) deveibped i n P a r t 3« Unft»r%uhately I t does hot seem p o s s i b l e t o Use the compactness o f G^k^ alone t o construct the^ e i g e h v a l u e s a j i d eigeriftihLCtibnS; Even though brie can show the exist e n c e of X n and u (x) such t h a t Au^(x) + X„uY(x) = 0, I t does not Seem n v n* ' n n".../-. ' p o s s i b l e i t o sho^ t h a t u n ( x ) I s - z e r o on the boundary, and thereby avoid the c a l c u l a t i o n s i n P a r t 3 . . Using Theorem 3.7. (Green's f u n c t i o n inversion) and the d e f i n i t i o n of G ^ (x,y, X), one can e a s i l y see t h a t ...c:a<^ i;v,x).ii u n) - ( v ; > r\f?'); > where {u n(x)} arid ( X n ) are:as constructed i n P a r t 3• Combining 51. t h i s with Theorem 4.2 we can prove the fallowing re suit on the eigenvalues. -2k Theorem 4 . 3 . The series E X n converges; i f k:> k^, where k Q i S ; i d e f i n e d b y the remark fblloWirig Theorem 4 .2 . Proof. For any set; of orthonormal functions/ f o r example {u n(x) }, the Be'ssel i n e q u a l i t y h o l d s ; i . e i cb I I G n l 2 < Mi) II2 :where ; c n = ( f , u R ) . n=0 By the inversion r e s u l t f o r we have ; d n ( X n - X ) * k u ^ y ) when f(x) = <slk)(x,y,X) . Thus If or k = k^ . o N | I X n - X | - 2 k u n 2 ( y ) < jl0^> (> ,y,X)||2 , n=0 where N Is as large as we please. Integrating t h i s expression with respect to y we get N I I X n - X l ~ 2 k < J* J |G( k>(x,y ,X )I 2dx dy, n=0 E E which Is f i n i t e by Theorem 4 .2 , i f X = : - « J 2 and to i s large enough, and: IS independent of N. Hence E [X^XI < «. Since X. oo, t h i s Implies that £ X_ converges, which completes n=0 n Theorem 4 . 3 . Theorem 4 .4. (The expansion formula). Let k k^, and l e t f (x) 52. have s u f f i c i e n t derivatives sb that the;fuhctibhs f , Af(x), A ( A f ( x ) ) , A ^ f (x) are a i l cbntihhbus, contained i n Lg(E).;•'> and tend to zero at theboundary of E. Then 00 f ( x ) =;;£ <?h where c r t fe;^,.^') . n=o Proof. Let c'^}-''m'y(t^\9 i u n) where f ^ ^ x ) » -Af(x) and f M ( x ) = - A f ( 1 " % x ) I > 1. By Giazman [7, Thebrem; 34, p. .90] we have; (Aui v) = (Av, u) where i u , e L g ( E j and both u arid v are zero bn aE;; Thus;if l±); . we set v = u n and: u = f v '••, we have 00 00 Let g(x) = ^ c n u n(x) = J c n ( k ) x n " k u n ( x ) • n=0 n=0 The method of prbbf w i l l be to show that (11) convergesand defines a function which i s i d e n t i c a l to f ( x ) . We f i r s t must show that f o r some constant C we have I (^2k + D"1 % 2 ( x ) < C (*) n=0 unifbrmly f o r x € E' As i n Thebrem 4.3 we have oo O n=0 5 3 . We have by Theorem 2.6 i f X e L(v)> and l i k e w i s e Since ||G(x, -,X)|| < K f p t x e E J and X e L Q , we have . oo . " .• . ' 0 I |X n- X | - 2 k u n 2 ( x ) <K. n=0 f o r a l l X € E v and X e L .. Since X„- », (*•) f o l l o w s . A _ o n 0 • •'"•'!.• We^ridw'^Qiisider the t a i l end,of the s e r i e s (11) . I f X N + 1 > 1, then by the Sdhwafz i n e q u a l i t y n=N+l n=N+l n=N+l 2k ok where a„ = (X» +1) X„ < constant. Note t h a t the s e r i e s n v n ' n — 2 [c ] 2 converges, being the " P o u r i e r " s e r i e s of which i s i n L 2 ( E ) by hypothesise I n view of (*.) and f ( k ) e L 2 ( E ) the t a i l end of the s e r i e s can be made small by the choice o f N , uni f o r m l y f o r x e E i . Hence s e r i e s (11) d e f i n e s a continuous f u n c t i o n i n s i d e E. A l s o n=o n=o from which i t f o l l o w s CO 1st*)!2 <. I "°n<K))2 ' I ' n=o n=0 54. which i s bounded by Theorem 4.3 (note k = k 0 ) . A s i m i l a r argument shows t h a t m n=o epiiyef ges,-'in;:'mean. square * to g(x) over E. Hence, by the Schwarz I n e q u a l i t y , ' f o r any n l i m ( u n , g-gj = 0 . m -» eo However we know t h a t (u , g_) = 'e . (m > n ) , from which i t f o l l o w s _ , v n m' n x — ' th a t ( u n , g) = c n . Thus the f u n c t i o n f ( x ) - g(x) i s an L 2 ( E ) -f u n c t i o n , a l l of whose "Fourier'yp.e.fficI.entsV.Ivajiish;;-- Hence by the P a r s e v a l Theorem Jjf - g|| = 0. Since t h e .integrand i s /contin-uous i n s i d e E, i t must vanish everywhere; i h s i d e E ; : T h i s cbmpiletes; the .proof of Theorem 4:4^  We; aref how. i n a p o s i t i o n : t p prove the main r e s u l t i n t h i s : p a r t , namely t h a t G l k^x>y,X) tends t o zero as x -• dE. The, proof w i l l depend d i r e c t l y bh the^ kno^edge/that G^ k) i s \ H l l b e r t - S c h m i d t and thus the s e r i e s E X^ cohvergent. The method i s ^ d e r i v e d from Titchmarsh [10> p. 106]. Theorem 4.5. For f i x e d x € E and f i x e d X e C not oh the noh-hegative : r e a l a x i s , G ^ ( x , y , X ) tends "to zero as y approaches the boundary Of E. (k > k Q + 2 and dim Ei= 2*) Proof. L e t r = |x-u| and d e f i n e 55. F(x,u) = - ^  l o g ~ g(r) , where R i s such th a t the c i r c l e |x-u| < R i s i n s i d e E f o r f i x e d U j and g ( r ) has the f o l l o w i n g p r o p e r t i e s : g e G*(E) , g(r) = 1 r < R/2 and g(r) = 0 T..> R . Since G ^ " i s not s i n g u l a r , we have i n the Green's formula.for G^k^(x,y,X.) and P(x,u) as f u n c t i o n s of x : J. ( G ( k ) ( x , y , x ) AP(x,u) - P(x,u) A G ^ (x,y,X) )dx r<R / k\ r=R' the s i n g u l a r i t y of P(x,u) a t x = u g i v i n g r i s e to the term G^ k^(u,y,X). Consider the boundary term: P ( r ) = 0 f o r r = R, and 11 - |Z = o f o r r = R, by the d e f i n i t i o n of P. Thus the an ar boundary term of (12) vanishes. Upon s u b s t i t u t i o n of AG ( k ) ( x , y , \ ) = -\ G ( k ) ( x , y , \ ) - G^" 1* (x,y,X) ( k > l ) , (k) which f o l l o w s d i r e c t l y from the d e f i n i t i o n of Gv ' and Theorem 2.6, i n t o (12), we have G ( k ) ( x , y , X ) = J [AP(x,u) + XP(x,u)] o( k )(x,y,X) dx f<R + J p(x,u) . o^"^(x J y A } ' d x - " C1?). r<R 09 R e c a l l the r e s u l t n ? 0 a n b n = where a n = (f»u.n) a n d 56. t>n = (&) ^ n ) , which followed d i r e c t l y from the Parseval Theorem. Let d n(u) = (AF(;,u) + XF(-,u), u n) and d n(u) - (F(-,u), u n ) , and r e c a l l that (G^(' >7,\), u n ) » (*n- X ) " k u R(y) . Applying these r e s u l t s to (13) we get CO G ( k )(u,y,X) = £ d n(u) (X n- X ) " k u n ( y ) n=o cs ,+ I d n(u) (X n- X ) " k + 1 u n ( y ) . (14) n=o We want to show that G^k^(u,y,X) goes to zero as y - aE. Consider the second series i n (14); a s i m i l a r r e s u l t w i l l apply t o the f i r s t . By Lemma 3-9 we have |u n(y)| < K X n f o r y near to the boundary; F i r s t consider the t a i l end of the series on the r i g h t hand side of (14), namely n=N n=N - n=N n*o ni=N < K . i i P ( - , u ) i i - ( | x ; 2 k + * ) 1 / 2 , n=N since X Now by hypothesis k > k Q + 2 and so the series . _ Pk+4 S.X converges. Thus given any £ > € > 0 we can f i n d an N so large that | I d n ( u ) (X n- X ) " k + 1 u n(y) n=N 57. u n i f o r m l y i n y, y i n a neighbourhood of the boundary. I f y i s s u f f i e i e n t l y c l o s e t o the boundary of E we have |d n(u) ( \ n - V X ) 1 * u n ( y ) | < 672N . f o r n=0, 1, . . N - 1 , since u and X are f i x e d ^nd u Q ( y ) tends to zero f o r each n. Combining a l l these estimates we get * N-1 | Xa n(u) (X n-Vxr** u n(y)| < I I d„(u) ( X n - X ) " k + 1 u n(y) n=o n=o n=N < N( £/2N) + €/2 = £ . Thus G^ K)(x,y,X) goes t o zero as y tends t o the boundary of E, when x and X are f i x e d . We can prove the f o l l o w i n g theorems i n a . s i m i l a r manner. -Theorem 4.6. The f u n c t i o n H(x,X,f) = T(G(X, •, X) , f ) s a t i s f i e s the equation AH + XH = f and the boundary c o n d i t i o n H ( x , X , f ) - 0 as x - SE\, as long as f , A f ' , A ( A f ) , ; , . , A ^ f are a l l c o n t i n -uous, contained i n L 2 ( E ) and tend t o zero at the boundary of E, where k = k . X i s a complex number not on the non-negative o, . x r e a l a x i s . Proof. H(x,X,f) S a t i s f i e s the equation AH -f XH = f by Theorem 58. 2.6. By the d e f i n i t i o n of H(x,X,f) we have 00 H(x,X,f) = £ c n ( x - X n ) _ 1 u n ( x ) , (15) ri=o where c n 5 3 (f> u n ) b y the Pars e v a l Theorem. As ;we showed i n t h e p r b o f . o f t he expansion theorem (Theorem 4.4) c n ^ » X n k c n > . where E - [ ' . c n ^ ) : ] 2 < * . Thus, s u b s t i t u t i n g t h i s r e s u l t i n t o (15), we get 00 H(x,X.,f) = I cn<*> X n" k (X - X ^ " 1 u n ( x ) . n=o •. We now ap^ply the same a n a l y s i s t o t h i s equation as we d i d t o equation (14) i n the previous theorem, t o conclude t h a t H(x,X,f) goes t o zero on the boundary. Note: k - k w i l l be s u f f i c i e n t to c a r r y out the c a l c u l a t i o n s . - 2 k Remark: Note that Theorem 4 .3 (2 7^ < *) holds f o r a domain E which l i e s i n the'half-space G and I s bounded by the sur-f a c e obtained by r o t a t i n g a t p * ( x 1 ) , where cp* s a t i s f i e s the co n d i t i o n s of cp, about the x 1 - a x i s . This f o l l o w s from the f a c t t h a t the. eigenvalues of E dominate the eigenvalues of the sur-face of r e v o l u t i o n (see Glazman [7, p.229]). Furthermore Theorem 4.3 now Implies t h a t the other theorems of:the s e c t i o n , namely, Theorem 4 . 4 , 4.5 and 4.6 hold f o r such a domain. Some m b a i f l c a t i b n s are necessary f o r example the s i n g u l a r i t y of the function..',P(x,u) i n Theorem 4 . 5 . 59. P a r t 5. Boundary behaviour arid uniqueness of the Green's F u n c t i o n . We are now i n a p o s i t i o n t o prove t h a t the Green's f u n c t i o n tends t o zero at: the boundary bf E. We continue t o make the assumptions of P a r t 4, so t h a t some i t e r a t e G^ k)(x,y,X) s a t i s f i e s Theorem 4 . 5 . Theorem 5.1. For X = -uu , co > 0 and x e E, G(x,y, x) tends to zero as y tends t o the boundary "of E. Proof. The proof i s performed step by step. One proceeds as f o l l o w s : Suppose f o r example th a t G ^ ) ( X j y , X ) - 0 as y - 3E Let us assume ( X Q i s negative) that G ^ 2 ^ ( x , y , X Q ) / 0 as y - BE . Thus there e x i s t s a 6 and a sequence y n -» z e 3E such t h a t ^ 2 ^ ( x , y n , X 0 ) > 6 > 0 f o r a l l ri ( l ) and l * - y n l > a > 0 , R e c a l l the estimate ( s i m i l a r r e s u l t s hold i t e r a t e s and dimensions) f o r dim E = 2: 0 < G^2^ (!x,y,-u)2) < M exp(-u>| x-y |/2) . 6 0 . Thus we have 0 < 2 ) (x,yn>-u>2) ...C;M exp( -t»| x-y n| /2) < M exp(-ij) a/2) ' — a where M i s a constant independent of oo (see equation 4 .6 of Thebrem 4.1)". From t h i s i t f o l l o w s t h a t there e x i s t s a o ' '. • . . . . . . i X - ^ s -oo^ ) such t h a t 0 < G ^ ^ X ^ A - L ) < 6/4 f o r a l l n. (2) By thebrem 2.10 we have D x G ( 2 ),(x,y,X) = G ( 5 )(x,y,X) i f X i s neg a t i v e , from which I t f o l l o w s t h a t ' X l G^UyT^) - G ( 2 ) ( x , y n , X 0 ) = J G ^ . ( x , y n , s ) d s . (3) Since the proof of Theorem 4.5 c l e a r l y shows th a t 0.^^-(3c,yn,X) - 0 as y R - z e aE u n i f o r m l y f o r X-^  s <_ X Q < 0, we have " • \ . l i m [ G ^ ^ ( x , y .s)ds = 0 . Thus from (j>) i t f o l l o w s t h a t as n -»*:» i G ^ ^ x ^ X ^ - G ( 2 > ( x , y n , X 0 ) | -..0, (4) which c o n t r a d i c t s equations (1) and ( 2 ) . Thus / ^ ( x , y , \ ) tends 6 l . t o zero, as y 3E, f o r each x and negative X. To prove the f i n a l r e s u l t we use the following'argument: By Theorem 2.10 and the same argument used t o prove equation (4) i t f o l l o w s ; t h a t |G(x,y n,X 1) - G(x,y n,X Q) - (X]_- X 6) G ( 2 ) ( k , y n , X Q ) 1 - 0 as" . n. •-» ». The argument proceeds i n e x a c t l y the Same manner, except f o r the i n t r o d u c t i o n of the term '.••"(• X^ "-''XQ) G^ 2^(x,y n,X Q) Which we already know goes t o zero as n .-''»'. For higher values of k the p r o c e d u r e i i s c l e a r , t h a t i s y reduce the r e s u l t one step at a time u n t i l one has G(x,y,X) -* 0 as y -» BE. Theorem 5.2. The Green's f u n c t i o n G(xyy,X): i s - u n i q u e f o r E i f X i s negative. ";•-. Proof. L e t Gj- and G 2 he two Green's f a c t i o n s f o r the same negative X and set f ( x , X) = G 1(x,y,X) -^^i7^})^y^:i(<^^,) • By the r e s u l t s of P a r t 2 we have (A + X) f(x,X) - 0 x e E-i f X i s not on the non-negative r e a l a x i s . I f X i s r e a l and negative we can show f(x,X) s 0, hy the maximum p r i n c i p l e . The maximum-principle s t a t e s t h a t I f AU(X) + a(x) - u(x) = 0 f o r x i n a bounded opein set K , i f a(x) <_ 0 on X, and i f u(x) 62. a t t a i n s i t s maximum at an';interior"1 pbiiit''bf '"-K,..^  then' u(x) = constant on K . To apply t h i s t o f ( x , x ) suppose t h a t f(x,X) j£ 0 on E, so, without Toss of g e n e r a l i t y , there i s a p o i n t x Q e E w i t h f ( x Q , X ) = a > 0. Let K = E^ where X I s chosen so l a r g e t h a t ( c f . Theorem 2.6) x Q € E^ and | f ( x , X ) | < a/2 f o r |x| > X. Then | f ( x , X ) | < a f o r any x e 9K, f ( x Q , X ) = oy and t h e r e f o r e f(x,X) must:achieve i t s maxlmttm at an I n t e r i o r p o i n t o f X. Consequently f(x,X) s constant oh K, arid thus f(x,X) '.•= 0 on K = E x , hy Theorem 5.1. Since X may he chosen a r h i t r a r l l y l a r g e , f(x,X) = 0 on E. 63. P a r t 6. A p p l i c a t i o n s of the Green 1s:Function. I n t h i s p a r t we S h a l l prove two asymptotic 1 p r o p e r t i e s , one f o r the eigenvalues and one f o r the eigenfunctidris as defined In;;part: 3. Throughout th$s p a r t we s h a l l assume'that E s a t i s f i e s a l l the c o n d i t i o n s Of the prevlbusitheorems i n order t h a t we s h a l l have a •": Green;'?'sif u n c t i o n w h i c h , i s 'zero ion t h e l h o M d a r y of E. Furthermbrey we:;assume t h a t the dimehsibh of E v i s 2. L e t F(x) ^Mx - G(x,y,X) ( 1 ) where p = |x-y[. The f i r s t term on the r i g h t hand side I s the Green's f u n c t i o n f o r A I n the iwhole^pl at x=y: Since a l l Green's f u n c t i o n s have the same'type of s i n g u l a r i t y as p -• 0. ., OUr f i r s t task i s t o o b t a i n hounds and asymptotic e s t i m a t e s " f o r the f u n c t i o n F(x,y,-u) and I t s : d e r l y a t l v e s w i t h respect t o u. I f we set X = - j i where u I s r e a l and p o s i t i v e , then AF(x) = u F(x) . Now we wish t o apply the maximum p r i n c i p l e t o F ( x ) : however, since E I s not bounded we need t o know t h a t F.(x) • •-•0 as x'•"-». •. By Theorem 2.8 we have t h a t G(x) .-» 0, as x » f o r f i x e d X and y. Furthermore H ^ goes t o zero as: p so we have 64. the required r e s u l t that F(x) -* 0 as x -• «. Now i f we apply the maximum p r i n c i p l e i n the usual manner (see Titchmarsh [10, p.169] we have that F(x) must assume i t s maximum on the boundary of E. I f x '.is i n t e r i o r to E and y i s on the boundary, G(x,y,-u) = 0 , and so F(x,y,- U) = | i . H ^ 1 ) (pj=fl - ( 2 r r ) _ 1 K Q( pjfl i n the usual notations of Bessel functions; Now K v (t) i s a o x ' p o s i t i v e , s t r i c t l y decreasing function of t . Hence the r i g h t hand side l i e s between 0 and (2ir)~^~ K Q ( a ^ ) , where a i s the distance from x to=the nearest point oh the boundary. I f we now regard x as f i x e d and y varying, we obtain 0 < F(x,y,- U) < : ( 2 T T ) _ 1 KQ(aS) (2) f o r a l l x and y. Since P Is, continuous at x=*y t h i s i s true f o r x=y. I f we again apply the r e s u l t s i n Titchmarsh [10, p. 170] we can show - max { ( 2 T T U ) " 1 KQ (aViI) , ( ^ u 1 / 2 ) " 1 a K^aJv) } < F^(x,y) < 0 where P,, = F. The next step i s to extend these r e s u l t s to higher d e r i v a t i v e s . 65. Let F J = D^F. From Titchmarsh [IG, ,-p\l-70] we know pl. u( x>:y>-u) = - % u ~ l / 2 K^p^u) y e BE . We wish to show' that F i r s t / b f • afl:/..we; can show u Kp<*/d = * J T - . k/2 " hy applying^ .the';'e;'4uati:phis ;f t i g t ) ; ^ - v ; ; K v f t ) - t K v_^(t) » v K v ( t ) - t K ^ C t ) . Equationsv (4) can be, found i n WatsOn [ i i , / p . 7 9 Se'c. 3.71]. The next step Is/to d i f f e r e n t i a t e the l a s t term i n (1), but by Theorem 2.10 D k G(x,y,X) = k! G ( k +^(x>y,X) , and thus f o r y e SE the deriVatlve Of the term G(x,y,-u) w i l l be zero since ^ Is r e a l and G^ k^(x,y,-a)= 0 f o r y e dE. This completes the 6ai;cuiati6n f o r equation (3) . ^\F^6m^"e^atid:h^••':(4) I t i s easy t o see that D t ( t k K k (t)) = - t k K ^ C t ) . However, /since K i s ; a! p o s i t i v e : f u n c t i o n , i t - f o l l o w s that t k K f c(t) Is a/positive" s t r i c t l y decreasing fMcMbh of t . Hence, 66. i f y € 3E, then a 2 K2^ a»/^ 0 < F 2 . M ( x , y , - M ) < ^ - ^ 7 — (5) where a i s the distance between x and the nearest boundary p o i n t . I n order, t o apply the maximum p r i n c i p l e t o F~ we must f i r s t show t h a t A p k . p : + k p ( k - i ) . M • <6> [(1.) 6 '• F . s a t i s f i e s the d i s t r i b u t i o n equation Since both H^ 1/ and G are.Green's f u n c t i o n s (see equation ( l ) ) , "A F = u F + 6 ' . We o b t a i n equation (6) by d i f f e r e n t i a t i r i g the d i s t r i b u t i o n equation k-times w i t h respect t o u, and observing t h a t D 8 0. As a s p e c i a l case of (6) we.have * P 2 - M = P P 2 ^ u + 2 P n ' <7> We s h a l l work w i t h (7) i n order t o o b t a i n a bound f o r F p . us i n g the bounds already known f o r F and FY Higher r e s u l t s can be shown by i n d u c t i o n . Suppose T?2'\x n a s a n e g a t i v e minimum i n s i d e E (note F p < > 0 Ort dE and i t tends t o zero f o r l a r g e x ) . Therefore 67. but t h i s c o n t r a d i c t s e q u a t i o n (7) ,• so Fg.^ i s non^negative on E. Suppose next that Pg . takes on a value, i n s i d e E, grea t e r than the r i g h t hand side of equation (5), then Fg.^ must have a p o s i t i v e maximum i n s i d e E. i . e . and hence o o „ K (a^ii) a K, (a,/u) F 0 < - - F < max - { — — , ^ , 0 } . 2 , u - U U ~ M ^ % u 1 / 2 Thus we have K (ayu) a K,(a,/u) a 2 Kg(ay u) 0 < F p (x,y,-u) < max{— A , — * • — r " — ' 3 » M TT u 2TT 8 TT U By an i n d u c t i v e argument we have k , a 1 K. (a^u) F k ( x , y , - u ) < max {-—-. i+1 k-i/2 5' * M 1=0,1,...k 1! 2 1 + i u K 1 / d We are now i n a p o s i t i o n to extend the asymptotic formula f o r the eigenvalues^: given by C l a r k I n [3], t o Our domain .E. Let T(X) be any f u n c t i o n such t h a t 0 < T(X) < A a ( x ) k ' + < £ : , where A and £ are p o s i t i v e constants; a(x) i s the dis t a n c e between x and the nearest p o i n t on the boundary, and k i s a -integer such t h a t t h e - I n t e g r a l of a ( x ) k ' over E i s f i n i t e and J\ J* [ G ^ (x,y,X) | 2dy dx < « where m = [ ^ ] E E Is also f i n i t e . . Furthermore.let 6 8 . { 0 x < Xo p where T Q = ( T , |u n| ), u n "being the usual eigenf unctions defined In Part 3 , Theorem 6 . 1 . I f dim E = 2, and N T(X) and T(X) are as defined above, then f T ( X ) - ^ J T W dx . E Proof. I f X and X' are not eigenvalues, then £ u / x ) u (y) (X - X») G(x,y,X) - G(x,y,X') = ) f -n=o ( V X) (V X') by the r e s u l t s of the previous sections. I f we l e t X = -u and X' = -u', where |JL and | i f are p o s i t i v e , then 1% u_(x) u (y) , _(„. - M) I n n = | [K (pyn) - K Q ( p ^ . ) ] n=o ( ^ ( V ^ ) - F(x,y,-u) + F(x,y,-u') . I f we divide both sides of t h i s expression by u' - u and l e t fi' tend to u, we have ( I f we can show the series converges uniformly with respect to u') j Z u (x) u (y) , n=o ^ *n + W 69. To show the se^eB^'d6^verge's''un±tp^lj'^ consider the t a i l end, of the s e r i e s : n=N C*riH*1>' " Now since^ tends t o u, -u 1 can be contained i n a set L Q (see P a r t 2), and thus » 2/ \ u„(y) by Lemma 3.5 and Theorem 2.5 (note y I s f i x e d ) , where K i s independent of y and [x\ . Thus the s e r i e s i s . u n i f o r m l y convergent, since the t a i l end can be made small (by choice of N) indepen-dent of u' . Since p K1(p^u) H• °- • • • 2 ir u and K ^ ( t ) - ^ s t - 1 as t -* 0, we have » / \ 2 •rv u fx) , by l e t t i n g y -• x i n expression ( 8 ) , i . e . p 0. We prove t h a t the s e r i e s i s u n i f o r m l y convergent i n y by a methbd s i m i l a r to t h a t which was used before. The^nextIstep i s t o d i f f e r e n t i a t e the expression (9) t o 70. b u i l d up the f o l l o w i n g set of equations: n=o n -I n order t o pass the d e r i v a t i v e through the summation s i g n we must show t h a t the r e s u l t i n g s e r i e s i s u n i f o r m l y convergent w i t h respect to the v a r i a b l e i n question. Again consider the t a i l end of a Serle s 1Ike those i n (10); i . e . « 2/ V u n ( 3 "2" where k i s a f r e e Index g r e a t e r than zero. Thus as u s u a l I , ^ r + 2 < 1 k l lo .( * ;s-u)H < V 1 K > w U + n ) (x +u) n n=Nv ri™' v A n where , K I s bounded, since x i s f i x e d and -y e L Q . Thus the s e r i e s I n (lO) Is'• u n i f o r m l y convergent i n u f o r any 'k> 2, "and; hence;we ban d i f f e r e h t l a t e (9) w i t h respect t o p as many times as we pl e a s e . L e t T ( X ) be defined as i n the statement of t h i s theorem. M u l t i p l y expression (10) by T ( X ) and I n t e g r a t e j t h i s g i v e s ( i f we can pass the I n t e g r a l s i g n through the summation) £ 1* T ( X ) U 2 ( X ) I P , , + ^ k ¥ - J t(x> F k - ( i ( x ' x ) d x - ( i : L ) E ;• We s h a l l show d i r e c t l y t h a t the i n t e g r a l and summation si g n i n ( l l ) can be interchanged. L e t 71 • .. 2 n=o< Consider ,T 0 u n 2 ( x ) dx eo _ U_2, E n=N 0 9 2/ A _T <• U„ (x) dx E n = o u n + | i ' oo ^ 2^x^ since T(X) i s e v e n t u a l l y l e s s than one. I f we l e t m = [ j r * ] * ' then the t a i l end! i s l e s s than K X" 1 J J |G^)(x,,yvX)| 2dy dx E E Which I s bounded hy the hypothesis on k'. Thus we can make the o r i g i n a l d i f f e r e n c e as small as we please by a s u f f i c i e n t l y l a r g e choice of N, and hence we can pass the i n t e g r a t i o n s i g h through the summation s i g n i n (11) f o r k >. k'. I f we use the d e f i n i t i o n of T^., namely % = ( T> u n)> the expression on the l e f t hand side of (11) can be expressed as CO I ( X n + u ) - k - 1 T n . n=o We wish t o express t h i s t s e r i e s as an I n t e g r a l i n order t o apply 7 72. a Tauberian theorem. Consider N T ( X ) * f -HXP *X - 7 | -/ D \ . A (T- + ... + TJ) 0 ( ^ 2 j t o i 3 . + 0 6 k+l Z. , ,•: . \ fc+i d=o ( X J + ^ Thus by combining t h i s r e s u l t w i t h (11) we have P N ( X ) k 1 r J , -Tk+2 d X• " ( 4 7 r k ( k + 1 >tf> J t ( x ) d x Q(X+ U ) E + f T ( x ) F. (x,x)dx. (12) ( k + l ) ! E ^ Le t us assume f o r the moment tha t J T(X) P ^ . ^ x ^ x j d x = 0 ( | i - k " ^ ) (u - •) (13) E where C i s someL f i x e d ; p o s i t i v e number. T h i s r e s u l t w i l l be proved a f t e r the a p p l i c a t i o n - o f the f b l i b W i n g Tauberian theorem t o expression (12), (see Titchmarsh [10, p.364]) . -;;lf. 0 (x+y)" x^ where a > 1 and 0 < $ < a, then f ( x ) c r ( g ) x " - 6 " 1 . P(a-p)rO) Thus-:we have from equation (12) and the h y p o t h e s i s , equation (13), 7 3 . t h a t N (\) ~ - T ^ + 2 ) _ J L _ f T ( x ) d x . r (2 ) T(k) 4Tr(k+l)k J . Hence N j X ) ~. T.f(x)d±--,:  w E which i s what we wished t o prove. The theorem w i l l be complete i f we can show the hypothesis ( 1 3 ) . R e c a l l the c o n d i t i o n on T(X), namely 0 < T(X) ..<: A a ( x ) k , + < f k' where A and £ are constants and k' i s such t h a t a(x) i s i n t e g r a b l e over E, where a(x) i s the di s t a n c e between x and the nearest p o i n t o n the boundary. I f we apply the estimate •1/2 . K v ( t ) ~ (-gf) e" ' as t - co , which i s v a l i d f o r a l l v (see Watson [ 1 1 , sec. 7 . 2 3 ) , t o the bound we found f o r P ^ ^ ^ y V - i J i ) , we see t h a t f o r a^/jl >. 1 /2 , < C u ~ 5 / 4 a ? / 2 e x p ( - V l i ) Since the l a s t term i n the parentheses dominates the ot h e r s . The general case f o r F k < ^ ( x , x ) g i v e s | P l r. ( x , x ) | < CVO 2** 1)/ 4 a k " 1 / 2 e - a ^ 74. where C i s a constant and a^u 2 , 1 / 2 . Next we wish to examine |P. (x,x) | f o r a^/ji <_ 1/2. To; do t h i s we use the estimates K Q ( t ) 0( [ l o g t|) as t - 0 and K ± ( t ) = 0(t'^) as t 0 ( i > 1). T h e f i r s t t e r m I n the i n e q u a l i t y f o r P^; •(x,x) g i v e s r i s e t o K*U. . . K J a,/u) v ° k : :•••.•"..<; C u " K | l o g a ^ u l f o r a ^ j K 1/2. u The I * 1 ! term ( i > 0) g i v e s a i K i ( a ^ > , . a 1 . 1 ^ _ c -k k-i/2 4 C I i/2 k-i/2 " C ^ f o r a^ /u < 1/2. Thus the f i r s t term dominates and we have | F k i ) i ( x , x ) | < C u"fc | l o g aVu| f o r a^u < 1/2 . We s h a l l how attempt to estimate the I n t e g r a l i n equation (13). L e t E„ =» {x € E m a(x),/u <_ 2m~1} m = o, 1, 2, ..... I f we use the hypothesis on T(x) and the above estimate which a p p l i e s t o E Q (a(x),/u <_ 1/2), then I J T(X) F ^ x ^ d x < C | a ( x ) k + £ u " k | l o g a(x ) V u l dx o o 75-which i s l e s s than C u ^ j ] a ( x ) k + C | l o g a ( x ) | d X + | J|log > | a ( * ) k + £ - d x j . C-E E r t J o o Consider the f i r s t i n t e g r a l : J a ( x ) k + £ |log a ( x ) | d x < C; I a ( x ) k + < f / 2 d x < C max a(x) ^  f a ( x ) k dx, x e E. E. -1/2 hut f o r x e E a(x) and Hence the maximum over E of a(x) i s ; l e s s than & C/k A l s o | [ l o g u | a ( x ) k + £ : dx E '' IS: l e s s than G». p.~€^. P u t t i n g these together we have I | T(X); ,(x,x)dx < G u " k ~ ^ as |i >••«•; Consider next the i n t e g r a l oyer E m + 1 - E m f o r m=0, 1, 2y,.. . I.e. E , -E m+1 m C G „ - ( 2 k+l ) / 4 | a ( x ) k + £ a ( x ) f c - l / 2 e - a ( x ) ^ u ^  ; Em+1-Em We'' know from the d e f i n i t i o n of E m , t h a t i f ^/€ E ^ - E ^ then 76 and so our i n t e g r a l over E m + i ~ E m Is l e s s than G M - ( 2 k + 1 )A m a x { a ( x ) k + £ " e ^ * ^ } J a(x) kdx, x € Em+1" Em E -(2k*l)/4 { ( ^ H + £ - l / 2 exp(-2 n l- : L) ) J a(x) kdx which i n turn Is l e s s than C since a(x) < u " 1 / 2 2 m and a ( x ) ^ u > 2 m~ 1. Therefore I J T(x) F ^ x ^ l C j C * ; ^ ^ - l / % p ( - 2 a Em+1" Em but since 0 9 we have J T P k . dx < .C- w"k-f/2 i ^ ^ ^ ^ ' e x p t - ^ 1 ) , E-E m=o . o where the I n f i n i t e series c l e a r l y converges. Thus -k-£/2 J T Pk-u C j f u E-E o from which I t follows that J T Pk.u dx < C' (u + u ) E < C u""*"^ as u 77. The proof of the asymptotic formula i s now complete. Remark: Theorem 6.1 i s a generalisation of the w e l l known asymptotic formula f o r the eigenvalues (see Titchmarsh [10> p.172]). I f E i s a hounded set we can set T(X) => 1 throughout E. I f T(X) => 1, then TN = 1 f o r a l l n, from t h i s i t f o l l o w s t h a t N T(X) « N(X), where N(x) ; i s the number of eigenvalues l e s s than X. Thus f o r T(X) -= 1 Theorem 6.1 reduces t o N(X) ~ — area E. 4ir Theorem 6.2. I f then M X) =-X— + 0( X 1 / / 2 ) f o r each x. 4ir Proof. TJ s i n g the re S u i t s and n o t a t i o n • of Theorem 6.1, we have f o r dim E = 2 f - 1 , ( - i ) k p , » n t o ( X n + u ) k + 1 %k7 k i k ^ -Furthermore ioo oo 2 / V k+i n t ( ^ u ) k + 1 + I l i J - F - .„(x,x) . 4irk ( k + l ) | i * k i k v , k - uv 78. Thus, since F. n ( x , x ) < C exp(-a(x)yu) i f a(x)^/u > 1 where a(x) depends on x o n l y , we have J i l A L dx ~ 1 -•• - • h- as u - » . 0 ( X + u ) ' 47rfc(k+l) u I f we apply the Tauberian theorem used i n Theorem 6.1, then $(X) ~ — as X » . 4ir T h i s r e s u l t Can be imprbved t o •(x)- = f- + o(x1/2) by the: methods of Titchmarsh [10, p.198], 79. BIBLIOGRAPHY 1. SHMUEL AGMON, Lectures on e l l i p t i c boundary value problems, P r i n c e t o n , New Je r s e y , Van Nostrand (1965). 2. F .H. BROWNELL, Spectrum of the S t a t i c P o t e n t i a l Schrodinger equation over E n . Annals of Mathematics, Volume 54, No. 3. (Nov. 1951), P- 554-594. 3. COLIN W. CLARK, Ah asymptotic formula f o r the eigenvalues of the L a p l a e i a n Operator i n an unbounded domain, t o appear i n the B u l l , Amer. Math. Soc. 4. COLIN W. CLARK, An embedding theorem f o r f u n c t i o n spaces, t o appear i n the P a c i f i c J . Math. 5. R, COURANT AND D. HILBERT, Methods of Mathematical P h y s i c s , I and I I , I n t e r s c i e n c e : New York ( l953i 1962). 6. DUNFORD AND SCHWARTZ, L i n e a r Operators, I and I I , I n t e r s c i e n c e : New York (19 BT'y 1963) . 7- I.M. GLAZMAN, D i r e c t methods of q u a l i t a t i v e s p e c t r a l a n a l y s i s of s i n g u l a r d i f f e r e n t i a l operators. ( R u s s i a n ) , F i z m a t g i z : Moscow (1963). 8. A.M. MOLCANOV, On, the c o n d i t i o n s f o r d i s c r e t e n e s s of the spectrum of second order s e l f - a d j o i n t d i f f e r e n t i a l operators (Ru s s i a n ) , Trudy Mosk. Mat. Obshchestva 2 (1953), 169-200. 8o. 9. F. RELLICH, Das Eigenwertproblem of Au+Xu = 0 i n Halbro^reri^ i n , ''Essays;presented to R. Courant" .." The Courant Anniversary Volume (1948), p. 329-344. 10. E.G. TITCHMARSH, Eigenfunction Expansions Associated with Second Order D i f f e r e n t i a l Equations. Part I I , Oxford University Press: Oxford (1958). 11. G.N. WATSON, Theory of Bessel Functions Second E d i t i o n , Cambridge University Press:(1944)» 

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