On the Growth of the Eigenvalues of the Laplacian Operator in a Quasibounded Domain

COLIN CLARK

Communicated by M. M. SCHIrFER

1. Introduction

An unbounded open set t2 in Euclidean n-space R" is said to be quasibounded if the points xEt2 with I xl large are near the boundary 0f2 in the sense that

lim z(x) = 0 ,

where z (x) denotes the distance from x to a f2. Let L denote the L 2 (f2)-realization of the negative Laplacian, - A , with zero

boundary conditions. If f2 is quasibounded and satisfies some additional restrict- ions, then it is known [1, 4, 5, 9, 10] that L has compact resolvent, and consequently purely discrete spectrum, consisting of eigenvalues 2j satisfying

0<11__<22<... , 2 j ~ + o o as j ~ o o .

Concerning the growth of these eigenvalues, we have the following known results (under certain restrictions on O).

Theorem 1. Define the "trace function'" N(2) by

N(2)= E 1. 2j~_1

Then if f2 is a bounded or quasibounded open set in R", we have

(1) lira 2-"12N(2)=c,~,(0), p,(t2)< +oo n.--~ o0

where c. = r(21/~)" F ( i + n/2)]- '.

For the case of a bounded set, (1) was obtained by WEYL [11]. For an unbound- ed set of finite volume, it is due to GLAZMAN and SKACEK [7], and to CLARK [5]. The case of infinite volume is given by CLARK and HEWGILL [6].

Now let G, (x, y, 2) denote the Green's function corresponding to L + 21(2 < 0), this function has been investigated by HEWGILL [8]. The iterates G~ k) (x, y, 2) are defined inductively by setting G~l)= G~ and

G. (~+ 1)(x, y, 2)= ~ Gf.k)(x, z, 2) G.(z, y, 2) dz. $2

Eigenvalues of the Laplacian Operator 353

Theorem 2. (HEWGILL [8]). Suppose there exists a constant f l>0 such that

i~n(On{x:a<lxl<a+l))=O(a -p) a s a - ~ .

Then there exists an integer m, depending on n and fl, such that G(~ m) (x, y, 2) is a Hilbert-Schmidt kernel, i.e., G(, m) (x, y, 2)EL 2 (t2 X t2)for f i xed 2. Consequently

(2) lim sup 2- 2 m N (2) < o0.

Defining g(t2)=inf {ct: 2-~N(2) is bounded), we see that

(3) �89 n < g(t2)__< 2 m.

The upper estimate (2m) obtained by HEWGILL depends on how fast t2 shrinks at infinity, as measured by the parameter ft. In the present paper we prove the following result, which has the effect of replacing the fixed lower estimate �89 by an estimate (2k) which also depends on how fast O shrinks at infinity.

Theorem 3. Let 0 be a quasibounded open set in either 2 or 3 dimensions, and assume the existence of a Green's function Gn(x, y, 2) satisfying (8)- (12) below. Suppose that for a given integer k > 0 we have

(4) Sz(x) 4k-" dx diverges, fJ

where z(x)=dis t (x, at2). Then the iterated Green's kernel G~k)(x,y, 2) is not a Hilbert-Schmidt kernel. Consequently we have for any 8 > 0

(5) lim sup 2 - 2 k + g N ( 2 ) = dl- o o .

so that g(t2)>2k.

The restriction to n = 2 or 3 is for simplicity only; in fact the same results hold in any number of dimensions (naturally it is assumed that 4 k > n in any case).

For a discussion of the effect of the function ~ (x) on N(2) in the case of an elliptic operator acting in a bounded domain t2, see AGMON [2]. Note that for bounded domains, the effect of the boundary appears only in the error estimates for N(2). For quasibounded domains, however, the boundary affects the asymp- totic behaviour of N(2) itself.

From our examples (see (19) and (20) below), it appears that our lower estimate (2k) has roughly half the value of HEWGILL'S upper estimate (2m). In essence, the lower estimate takes into account only the boundary of t2, whereas the upper estimate takes the interior of t2 into account also. It would be interesting to be able to calculate g(t2) exactly for a single example of a region t2 of infinite measure.

2. The Green' Function

Let H.,o (r) denote the fundamental singularity for - A + o~ 2 in R". We have in particular [3]

(6) H2 ~,(r) = 4 H(~ (i a) r )= (D r

- 2 1 o g - ~ - - + O ( 1 ) as r ~ O +

and

(7) Ha,o(r) = 4 ~ r e-tO P.

354 C. CLARK:

For 4<0 the operator L + 2 I has a Green's function G~(x, y, 2), which satis- fies:

(8) Gn(x,y,2)=H~o,(lx-yl)+g~(x,y,2); ~2= - 4 ,

(9) Ay gn(x, y, 4 ) - 4 g,(x, y, 2 ) - 0 ,

(10) Gn(x,y, 2)=O ; x~I2, yEOt2,

(11) G~(x,y, 2)=G~(y,x, 2); x ,y~f2,

(12) G~(x,y, 2) oO; x~f2, y~oo(y~12).

Of course G~(x,y, 2) is a kernel for (L+2/) -1. These properties are not all obvious (especially (10)) in the quasibounded case, but they have been established, for certain types of sets f2, by HEW~ILL [8]. In this paper we will assume f2 to be such that G~(x, y, 2) exists and (8)-(12) are valid.

Lemma 1. There exists a constant c A > 0 such that

(13) G2(x,y,2)>_c ~ if I x -y l< �89

and

ca if I x -y l< �89 (14) Gs(x, y, 4)_>_ z(x)

Proof. Recall that ~(x)=dist (x, 0t2). Fix x~t2. It follows from (9) and (12) that +g~(x, y, 2) satisfies the maximum principle for y~f2. Consequently, H~o,(r) being monotone (cf. [3, formula (2.7)]), we have

- g~(x, y, 2) < max H~ ,o(I x - y 1) =H~,o (z (x)), y e O~j

so that

(15) G~(x, y, 2)_>_n~ o(I x - y l ) -n~ o(, (x)). To prove (13) it is sufficient to consider values of x outside some compact subset of D, i.e., values of x for which z(x) is small. For such x, (6) implies that for I x-y l <�89

[ _ ~ ] z(x) >Cl~lOg2=ca. c~ ~-log = q ~ l o g I x - y [ - G2(x,y ,2)>c~ - l o g 2

A similar calculation gives (14).

Lemma 2. For each positive integer k there exists a constant eka such that for n=2, 3:

(16) G~k)(x,y, 2)>=CkaZ(X) 2k-n if ly-xl__<�88

Proof. The case k = 1 is covered by the previous lemma. Suppose then that (16) holds for a given value of k; let n = 2. Since G2 (x, y, 2), and consequently also G~2 k) (x, y, 2), is non-negative on ~, we have

Gt2k+~)(x,Y,2) > I Gt2k)(x,z,2)G2(z,Y,2) dz Iz-xl ~-A ~ (x)

~CkACl2T'( x)2k-2 I dz Iz-xl<-~(x)

=Ck+I,A'C(N) 2k provided Ix-yl<=�88

Eigenvalues of the Laplacian Operator 355

where we have used the inequality G2(z, y, 2)>c1~, which is valid because

I z - Y l < ~ T(x) < �89

This proves (16) for n = 2 ; the case n = 3 is similar.

Proof of Theorem 3. If finite, the Hilbert-Schmidt norm of G(,, k) (x, y, 2) is equal to

(17) SS IG(,,k)(x,Y,2)12dxdy>cxS( ~ "c(x)'~k-2"dy)dx=c2~ "r(x)4k-'dx" rtxf~ ~ ly-xl<_~(x) s7

Hence (4) implies that G(~ k) is not a Hilbert-Schmidt kernel. This in turn implies oo

that y '2 j -2k diverges, and the remaining assertions follow immediately. 1

3. Comparison of the Estimates

Consider the plane region f2 defined by

(18) O<y<(x+l)- l /r ; x > 0 .

Accordingto[8, p. 160]wehaveg(f2)<=2['--22].SinceIz(x)Vdxdivergesifand f2

only if v____ ~ - 1, Theorem 3 above implies that

This is an explicit improved form of (3).

For the case of a tubular region I2 in 3-space, defined in terms of cylindrical coordinates by

O<r<(z+l)- l /2r; z > 0 ,

we obtain by the same calculations

(20) 2 [~---~-1 < g(f2) < 2 [ , + 1].

Both (19) and (20) exhibit a discrepancy, approximately by a factor of 2, in the lower and upper estimates. This discrepancy seems to be typical for our method. We remark that more precise estimates than (13) and (14) obviously hold for the Green's function; however the more precise estimates lead to the same estimates for the Hilbert-Schmidt norms of G (k) in terms of ~z(x)~dx.

In the case of n dimensions, inequality (16) becomes

G(k)(x,y, 2)> fCnk;~lx--yl2k-n~ if 2k<n (21)

=~Cnk2"~(X) 2k-n if 2 k ~ F l ,

provided Ix-yl<�88 Thus G (k) is never a Hilbert-Schmidt kernel (even for bounded f2) if 4k<n; if 4k>n, then (21) yields (17) and Theorem 3, as before.

This research was sponsored by the Air Force Office of Scientific Research, Office of Aero- space Research, United States Air Force, under AFOSR Grant No. AFOSR-68-1531.

356 C. CLARK" Eigenvalues of the Laplacian Operator

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The University of British Columbia Vancouver, British Columbia

(Received August 6, 1968)