Otemon Economic Studies, 17 （1984）

On asymptotic distribution of eigenvalues

for a certain elliptic operator in

a cylindrical domain

FUMIOKI ASAKURA

25

§0. In this note, we study the eigenvalue problem for the following elli-

ptic operator in the cylindrical domain R χ゙Ω⊂R“χ尺゛.

(0.1)　　　　Au ＝α(x, D 加 ＋q(x)b(y, DS)u　　(x ，3')∈だ ×Ω.

Here a(x, 既)is a second order elliptic operator in R" with C ゛-coe伍cients

of the form

(0.2) α(x, DこuこΣD,.(恥(^)D 声)　(d,= －v"二T二」

and b(y ， DS)is an elliptic operator in Ω⊂R ゛ which has a certain positive

self-adjoint realization in L(Ω)with or without boundary conditions.

If the spectrum of b(y, D こ)is discrete and o(x)satisfies

(0.3) a(j;)泡Oo＞0 ，lim Q(x) ＝・・，

then there eχist in μ(R' χΩ)a complete orthonormal system of eigenfunc-

tions {≪ } and corresponding eigenvalues {?.

} satisfying

(0.4)　　　　　　　　　λら＝λμ。in 犯×Ω.

Let N(t)・denote the number of eigenvalues not eχceeding t. We study the

asymptotic behavior of N(t)as t tends to infinity.

Since q(x) ろ(y， A)is regarded as an operator valued function of X, λ

is considered to be an elliptic (Schrodinger)operator with operator coe 伍-

cients. In this poin of view, Kostyuchenko-Levitan[7]and D. Robert 111]

have established the general methods of studying the eigenvalue problems

for these operators. Under certain conditions stated by the languages of

operator theory, they　obtained　the　asymptotic formula for iV().　H.

Tamura[12]adopted these methods for studying the eigenvalue problems

of the Laplacian in an　unbounded domain. F. Asakura[4]studied the

eigenvalue problem of the operator

(0.5)　　　　　Au ＝£沁＋q(え)D 沁　　in (0, ・・)×(0, ;r)

by means of the separation of variables.

In this note, we shall generalize some of the results in [4]to the

operator (0.1).Let {ω,(v)} be a complete orthonormal system of eigenfunc-

tions of b(y, フ:),)and {μj}corresponding eigenvalues. If we seek out the

(1)

26 FUMIOKI ASAKURA

eigenf unction of A in the form

(0.6)　　　　　　　　　　　 ゆk(ヱ，y) ＝(p(x)ωt(y)，

then g is an eigenfunction of the operator

(0.7) λ，タテぶ皿( 恥(え)£)即)十卸7(功　　 ん＝i，2 ，…).

We assume that we have the complete knowledge of the eigenvalues μ。for

example b(y ， A) ＝£? in (0, 7r)with the Dirichlet condition. Then our

considerations in this note consist in studying the eigenvalues of the operator

(0.6)with particular care on the parameter μt。

In　§2, we review the caluculus of pseudo-differential operators which

is the main tool in this note. Following the classical scheme. we construct

parametrices of y17 －λand compute the Hibert-Schmidt norm of these opera-

tors in　§3. In § 4, we study the zata function defined as

(0.8) GW ＝Σλご，

where λ，is the n-th. eigenvalue of A. In a similar manner we set

Uz) ＝Z μに

We assume that ら(;z)is absolutely convergent for Re z＞βo and that the

singularity of ぐb(z)with the largest real part appears on Re z ̂＝β. Let

“゜= inf

{“ei?;

∫／(^) ″^

倣det(“″(a:))"' 心 ＜ ゜゚

心＝max {αo，βo十昔}

・

Under these situations, we obtain the analytic continuation of C^(^)across

Re z ＝xo as the following

(0. 10) ぐぶ) ゛泉 吊 示Cb(

゛^/2爪

β(が/ ‾'(d^^(ら( え))' ″心

十Hz)

where r(z)is the Gamma function and h(z)is holomorphic in ・Re z＞x^ －δ.

By the Ikehara Tauberian theorems (see D.V. Widder[14]and also F.

Asakura[4]), we obtain the asymptotic form of iV().

Let 況()denote the number of eigenvalues of Aりnot e

χceeding t. For

each μt，considerations　of　these　operators (0.7)fall into　the　works　of

various authors (for example E. C. Titchmarsh[131, T.L. Pham[9] ，D ・

Robert[101).　D.　Robert[10] says

(0.11)　^ ≪ ＝匹 バ　　　　dx

考(1 十 〇(t-つ)with (0 ＜ 。＜ 生)

(2;r)　 ・・, f)＋J≫j9・く・　　　　　　　　　　　　　　　　　　　　4

In § 5, we show that the remainder estimate in （0.11) is valid uniformly

in k, by just following the arguments in ［10］and obtain the asymptotic

formula

( 2 ）

On asymptotic distribution of eigenvalues for a

certain elliptic operator in a cylindrical domain27

(0. 12)　2v(z)゚ てこア‾

ぶL。 dx 心(l十〇(r

つ)with ((0＜r＜^ '

This work was completed during my stay at the University of Paris-

Sud. I would like to e χpress my gratitude to professor J. M. Bony for his

great hospitality and incessant encouragements. The reseach tour is sup-

ported by the fund from Japan Private School Promotion Foundation and

Otemon-Gakuin University. l would like to thank those universities and

the foundation. Finally l would like to thank my colleagues in School of

Economics Otemon-Gakuin University who gave me the chance to make a

one-year reseach tour to Orsay.

§1.　Let us denote

28 FUMIOKI ASAKURA

ポ ＝(Σ 皿恥(ヱ)炳 十μq(x)).

■',y=iProposition 1. 2.　収A")is expressed as

(1. 10)

with

衣/l") ＝ α2．,。(^， ぐ 十α2．_u。:v， ぐ)

α2。。(^，ぐ)＝( Σaii(x) ξふ 十胆(x)) ’＝Φ人エ，ξ) ’

Moreover ^2m 、^)　a2ni-i、。satisfy

(1. 11)　　　　| ∂f£仙2．,バ:X, ξ ＼＜C 卵Φ。(x，ぐ^ ’"“ ，

(l. 12)　　　　| ∂f) グα,．-,．U ，ξ| ぶa, らU ，ξ^’i-i“!

・with constants C 。βindependent of pt・

Proof.　We show (1. 10), (1. 11)and (1. 12)by induction on ■m.

When m ＝l, (1. 10)is nothing more than (1. 9).　Using (1. 6), we can

readily see (1. 11)and (1. 12)for α2．and αi≫.If we assume that

(1. 10),

(1.11)and (1. 12)are correct for m. then by the Leibnitz formula, we find

″(ポ ¨サメ, 古

吋(″馬)) 舅(a( ポ))　

十

二らx ，ξ'″”^'十らm + l,衣ぶ，ξ・

We can readily verify that α2。iよsatisfies (1. 12)・

§2. In this section we shall review the calculus of pseudo-differential opera-

tors.　　Firstly we show a preliminary lemma.

Lemma 2.1.　// a positive C °°-function f(x)satisfies

(2.1)　　　| £ゴ( 功 くC 貨 〉^f( え)with a constant C,

then there exists a constant L such that

(2.2)　　　

2 ご 雪 雲 忠次官i乙 大詰 だrl 訟

Corollary 2. 2.　Assume （2.1 ）. Then it follows

29

(2.4)　　　　C

30 FUMIOKI ASAKURA

Following H.　Kumano-go[8]we define the oscillatory integral with

amplitude α(y, が)by

(2.11)C ≫5－∫ト・

゛α(y，η)め吻 ＝皆∫F・"・"

χ(ey, £rj)a(y, η)心 両・

Here 1 is a rapidly decreasing function satisfying χ(0, 0)＝l. If a(y, η)

has certain appropriate properties, the integral is defined independently of

the choice of 1. Using oscillatory integral, we find that P(x, D)is e χpres-

sed as

(2.12) 戸(x, 瓦) 衣功 ＝らバ レー

゛゙ '）(y，ηXoo 十y)dydr)，

which is valid for MG厩

Let 八，几亡召’・゚ and ぺ弓)＝ち(x ，ぐ). The product Pi 瓦is again a

pseudo-differential operator whose symbol is eχpressed as

(2.13)　a(P,P,)(x ，ξ ＝Os －＼＼e-‘'・'Pi(x，ξ十η)p2(x十y, ξ心 励・

Our calculus of pseudo-differential operator is based on the following lemma.

Lemma 2. 6. Let mj，ゆe W(R'xR")and ）e 名’ヾ 0" ＝l, 2). Then

(2.14) ら(^ レり ＝Os －∫^e・‘p・

？i(x，ξ十巧)p2(x 十ty, ミ 心 励

is ゛-function of (x，ξ)and satisfies

(2.15)　　　 臨,卵,。。くC ㈲ にIi,,。,＼p2＼l+I。。

where C and h depend only on d and L (L is the constant appearing in

(2. 6)).

Proof of the lemma is put off until Apendiχ. Later we shall make use

of the point that h depends only on j and L. Admitting this lemma, we

obtain the following theorem.

Theorem 2.7. Let ntj， Φ^W(R'XR")and p.e^ ”’,'゚Q ＝l，2)・

Then the product PIA is of class 名゙ ～゜and the symbol is expressed as

(2. 16) a(八八)＝Σ古 ∂ひ^(x，

ぐD か.(x，ξ 十りn(x，ξlaΓ＜Nα.

with rN^ 5"呻沙－“.Furthermore the following estimate holds

(2. 17) |な| l,・,。0- ″≦C＼夕l I /

+ /。。。- ″|戸2

に 。,。

where Co and I depend only on d and L.　　　　　　　　　十

Proof. 妖PiPi)is expressed as (2.13). Inserting into (2.13)the

Taylor eχpansion of Pi(x，ξ十η)

ブi( ゛ ぐ切 作 爪子 ∂

ひ.(Xバ が十I7l-万

・＼'s

卜 が ‾i∂ひte ぐ十drf)id,

we find that (2.16)holds with

(2. 18)

i/3|-

扁Jo

{05 づF ・゙ ∂訥(x ，ξ十向)£)涵( ヱ十y, ξdy 焉伺d．

Then employing Lemma 2.6, we obtain the theorem.

（6 ）

On asymptotic distribution of eigenvalues for a

certain elliptic operator in a cylindrical domain

31

Remark 2.8. Later we shall make use of the　class 6"'・゙ with の＝も

U ，ぐ.In this case we can see immediately that (2. 15)and (2.17)hold

uniformly in μ・|

In the last part of this section, we refer to the boundedness of pseudo-

differential operators. It is well known that ？gO ’・゙ is a bounded operator

in L( 沢?)if m is uniformly bounded. Here we consider certain conditions

under which ？ is of Hilbert-Schmidt class or trace class. Let 夕(x ，ぐ ∈

夕(R-'X だ)and u ∈ヌSet

尺(x, y) ＝ト

’-'か(x ，ξVf.

Then 尺(x ，y) ∈9'(R'' ×R")and Pu is eχpressed as

Pu(x') ＝∫K(x,

y)u(y)dy.

Denoting by ＼＼尺μthe Hilbert-Schmidt norm of K, we find by the Parseval

formula

＼＼K＼＼%s=[＼|K(x,

y)fdxdy

づ ド 専(x ，ξ)f右 虜

。 ＝jjl/'U ・Of 臨 砺レ

Then by limiting process, we obtain

Theorem 2. 9.　If 戸(ヱ，ξ)e 訃 ゜satisfies

lj 牡 丿戸(゙ ξ)卜O “nd

j戸u' ぐ ＼'右 府 ＜ ゜゚ '

Hilbert-Schmidt operator and ＼＼P＼＼hs is express

||パ ‰づしU

，ξf 右 府 。

(2. 19)

lj 牡 丿戸(゙ ξ)卜O “nd ＼M戸(x' ぐ ＼'右 府 ＜ ゜゚ '

then P is a Hilbert-Schmidt operator and ＼|P＼＼hs is expressed as

(2. 20)　　　 ＼|P＼‰＝[[ ＼p(x, ξf 右 府 。

We say that an operator K in £(尺“)is of trace class if K is eχpres-

sed as a product of Hilbert-Schmidt operators. The following theorem is

found in L. Hormander[61.

Theorem 2.10. If P(x, ξ ∈ざ1・゙ satisfies

(2. 21)　　　 Σ　jjlr ♂£ひU ，ξ ＼'心 応＜・・　for 2k ＜d,la十β＋7l

言 な な れ

Sc 完 二 れ ノ ご ご こ⊃ 六 万 二class

and

(2. 22)　　　∫卜 かU

バ 市 政 ＜・・( ≪ ＝0, 1)

holds, then Tr P is expressed as

(2. 23)　　　Tr P ＝＼＼p(x,

ξdx 政.

theo

ぷof. Since P(x, ξ)is integrable in (:r, ξ), we find by the Fubini

Pu(x) = 卜(x,

y')u(y)dy　( 尺(x ，y) ＝

レ(¨) 今 り ，ξdi) ．

こ 言 言 こ こ に

．

言 二 回 にT おas a trace x=y.

Tr ）＝匯(x

，sc')dx={{p(x, O 政dx.

§3. In this section we construct parametrices of A ≪　Firstly we construct

a left parametrix Ql. For simplicity we sometimes omit (1 to denote ゆ(x，ξ

＝ちx, ぐ ，A ＝A ．e.t.c.Following the classical scheme of construction

we set

(3.1)

UaビニT　　;i嘩(o,　ex）)

― ‾嶮し

ぶj‾ふ‾∂g鸚)f)7ら ＋1

巾尽-1子 ∂紅 偉 μ,

″J

q胃can be e χpressed in terms of derivatives of α2．and α2．_l as the following

i 2(i-*)十|≪l十|β|(3.2)　 ∂μ)グ㈹ ゛ぶ

ベ

ペ レ!]

川 岸

回y

二回|

C　∂μl' α,.…∂f)' α,．∂μ:y.'α,.-.…∂pD ？α2．-,

μ　　　　　　　　　　　(α.．－x)^ “-'

(see for example H. Kumano-go[8], Chap. 8].Since A" is a differential

operator, we note that the summation in I begins at [2mT

よ

We carry out some estimates for q 回.

(3.3)　　l ∂μ)グq胃(x ，ぐ|

＜C　j 2U-k)十|a|十|β1　　　　　　　　　　の(x，ξ)2’(“ ‾lvl‾l″|

‾ 尚乱=[ ペ レ

吐]川 順

護

二鴎りk2

≫u・ξ)－λげ'

＜C　　i 2(>-ん十【α|十|β1　ψ(x ，ぐ''(“) ‾'‾』“】

‾ ゛瓦

べ:j二j

レ!]丿 らU ・ ξ

‾卯“ ゛'

Magnitude of ＼ら －^卜s estimated from below as the following. Let d()

= dist (2,[0, oo])and d ＝a.Tg 2. We find

(8)

Hence we have

(3.4)

雲ご1 詐 謡 ご 訟 丿 器 忿1 昌j

‰ べ 巨印)

|ら づ

ソ サsin^ ド 背 ～(| ∂|+l + |“|∩

λΓ七鈴こ ヨ ル

Proof.　We have (3. 7)immediately. Using (3. 4)again, we obtain

(3. 8).N-1

Setting Q＝ΣQ 回with べQ 回)＝q回=0

theoremN-i　　　　　　　　　　　　　　　　N-1

べ亀U ”'－λ))－/＝Σ　 Σ斗 ∂釦回珂

～-1 十ΣH- 乱-i片0 |α|=.Vり-l α.　　　　　　　　　j=0

with

φ‰( ヱ，ぐ，X)

Iぶ

‾か ＼(i

‾∂)̈ {os づ 卜・゛∂W回り 心 十向}にな り 十y, ξ)右 府dd

(α－α2m ＋α2．+ l).

Since

|∂μ)グ(∂細野x ，ξ,二))

卜c郎(

☆)2j+|a|+| ″|+|7l+l斤1 畿釣％-｀-|“'-2’

we obtain by Lemma 2.6

Proposition 3. 2.　Set R む＝Q ム(A’－λ)－I and ぶ，:尺O ＝r:..　Then for

N ＞N 、，rL has the estimate

(3.9)　　 贈Z)配(^ ，ぐ，2) 匡a 。(五 万+i.+|a|+|/3|叶2m 心- ゛ ＼“＼

(9) ・

34　　　　　　　　　　　　　　　　FUMIOKI ASAKURA

where u depends only on d ， m and L.

In a similar fashion, we can construct a right parametri χ. As before.

we set

q謬こ　+1　 。　　λ申[0, a⊃万]，

(3.10)　　

Uニ レ し

乱 白|a|+;=/-lか

臨 一轟川

∧

0

On asymptotic distribution of eigenvalues for a

certain elliptic operator in a cylindrical domain35

Proof.　We note that the condition （1. 3）guarantees （3. 14）for large

辨. By Proposition 3.1 and 3.3, we have for q_j ＝q曾，q回

＼q-i(x，ξ，X) ＼£Cj(☆ ド1

Γ2"

36

(3. 23)

Furthermore

(3. 24)

FUMIOKI ASAKURA

||瓦|12 く/^( 言)4N N-2m-N, μd

‾ ( d _

防 ぷ ぶ 印 プ ニ誓ニヅ トイノ 臼)

ド ‰

holds for λe Σg and

(3. 25) ＼＼悩 ‰£C λΓ4″(応‾小り1 ‾j ／( 功1 ‾` 心

for λ∈柔（N, = 2十NJm not depending on N ）．

§4.　In this section　we　carry out the analytic continuation of the- zeta

function defined as (0.8). We may assume that the eigenfunction of A

has the form

のりx ，y) ＝ 卿/(ぶ)叫(y)

where 賎(y)is the eigenfunction of b (y, £,)corresponding to the k-th

eigenvalue, μt.　Let λ，be the eigenvalue corresponding to 貳( ぶ，v). Then

卿/satisfies

(4.1)d

Σ £・(a,7 (ぶ) £ 即u ＋ μtq (x)^h ・゛ 臨(p・i,ノ=i

We choose sufficiently large m, such that

(4.2)

holds and CbU － ^/2)＝

＼a^) 卜 句x＜oc

孟μご“/2 is absolutely convergent. Then we find

by Theorem 3. 5 that the parametrix Q of Am is a Hilbert-Schmidt opera-

tor. We denote by Q,() ＝λ( －λ)with λ＞o and μ＝μ,.We also denote

A,ニAれ.　We consider AT instead of

At.　Let us denote again by hi the

m-th power of λu．We sta rt with the identily

(AT 十λ)知＝(λ十臨)伽( λ＞^)・

Applying O,(yi)from the left side, we have

恥 十瓦(λ)恥＝(λ十臨)Q, ヌ)卿.

Then we obtain

(4.3)　　　　　　　　_ ヤヒ ＝仙(λ)り汁7 ことこ 瓦(2) 伽.

Taking the absolute value of (4.3), we find

(4.4) ＝|収( λ)川2 ＋2 み 罰≒7

瓦(λ)恥Q1( λ)恥

＋ |瓦()φP1

-λ＋ λZ

＝貼(λ)祠 十〇(λM貼 λ祠p) 十〇(rり瓦(λ)川D ，

where ^ ＝0（司means 晋is uniformly bounded.

（12 ）

2 ぶ 齢 昌 票ご 図f ぷ昌 治 。訟

Integrating the both sides of （4, 4）, we find

(4.5)ツ

≒ ≒臨() 川r 十〇(λN貼 λ)川P) 十〇(λ' 11 瓦 ン)祠D

゜

Summing up with respect to I, we have

(4.6)　　V ≒ 月

収(λ)‰十〇(λ'l貼 λ ‰)十〇(‾^R(λ)‰)゜

Since by (3.8)(with ＼a＼ = ＼β＼ ＝N,＝O)

祗)u ，ξ，べ)l< 十四( 功゛宍

＝よ‘( レ 迂回 二

戸耐(m'

＋i) ”網

－　r(mα－d/2) ＼一回　　　q(x)訃皿　j

(2π丿T (ma)μ　　

し(ゐz(恥(J)))1″｀2;

（by （4.2 ）the integral is convergent for α>l ）.

0n the other hand, we find

(4.9) Sil-“ln(2)|心

づ ン1-“-″2刈 汽 λ)||ば λ十(丁

λ'-“(l 十λ)'W㈹ λ)|臨心

£cr( プリ クラ1厩2 ゛) i‾̈ - 止∫／( 功

訃¨ 咄 ,

十Cμ ‾べ∫／ 功- ≒む)(∫ン ‾“(l ＋X)一失｀- dλ.

（13 ）

37

38

Denote

(4. 10)

For α之l ，

(4. 11)

Set

FUMIOKI ASAKURA

訃 ≒ 器 毎器 μ‾几ぺj

回 ぷ)μ・＋～) ．

ル私(α)|くcr( 胃⊇が4碩2m)r ( 十☆

‾づ)∫

／(功駈“

‾止血

＋C以 罵‾^/2)(

し呼丿 ″‾o血)(り1 ‾“(1り) ‾べ芹血

“'ぺAト; し(d 読に?ム;)1

が＆ぐ °}

βo ＝ini{β; ぐ。(β)is absolutely convergent}

a:o ＝max{ αo，β0十昔}

we can readuy see that (4 11) is uniformly convergent for α沿古zo

－ふ

十s

(s＞O).　Furthermore we find that

ど(z) ＝jk 仙(z)

is unijFormly convergent for zEGREz 泡士zo

一古

十sHence summing

up(4.9)in 1 ，wehave　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　■

(4. 12) ぃ( ・)＝ 諧 写 藻 汗( 一 寸 ん( 誤

回 ぶ)1

匹 十心)

where μ(z)is ho16morphic in Ra ＞古Jo

一古Changing the variable

ス こo 二

§5. In this section, we show the asymptotic formula for　N （t）with a

remainder estimate （0. 12）. By separation of variables, we consider again

the operator

(5.1)j

A,

≧ご 齢,に謡 忿i言言 訟 露l 訟

selves with showing the outline of the proof.

39

In § 3／we constructed the parametri χ Q 。 for y1;'.　Employing Theorem

2.10 together with the condition (1. 3), we find that Qh is of trace class

if we choose sufficiently large 。. With such m, we denote Q*( λ)＝Q 。Q)

with μ＝角.　Q ，satisfy

(5.2)　　　　　(A: －λ)Q,() ＝i－ 瓦 λ)

with remainder operator 瓦(X)which has the estimate (3.23). We can

readily see with large λ∈几 ＼(O ＜r ＜去)

防 心 几 ぶ{ ・　for all ん.　　　　　　　　　　・.

Then for such K ，At has the resolvent

(5.3)　　　　　(AT －x)' ＝Q,( λ)(i－瓦())' ・

Since Q* is of trace class, then it follows ( λΓ一ヌ)-l is also of trace class.

The resolvent is especially a Hibert-Schmidt operator and we find

(5.4)　　　　　|AT －λ ‰頑 ＼QA|i －R λλ)) 慨　　　　　　・■　・　．　　　　I

くと( ⑤)2 μj‾2’Lβ(J)1 ‾2’jj゛;

On the other hand, the resolvent is expressed as

(λΓ－λ)-'＝Q,( λ)十(λΓ- ヌ)- 瓦̂().

Then we find

＼Ti(ボ ーX)-'瓦( 掴 くU ぶ べ) 丿 副 瓦a) ‰

2N+1 _ N-2Ξニft, μ昔 ぺ しd ,

(we may assume No 之2m).

Hence we have

Proposition 5.1.

(5.5)　　|Trur －;i)二'

－Tr Q*() 匹ci λΓ叫か 小 ゛

μh

for λ^ 瓜where C and N, are independent of k, N.

Since the resolvent is a Hibert-Schmid operator, we find

(5.6)

Let us denote

－ΣrJ くoo.

1-1

iv*()＝゛{λ,;λ≪て拙

Then it follows from (5.6)

(5.7)　　　　　　　 況() ＝0()　as Z →oo.

Hence 篤(t'/つ＝O(f).　Then we may assume. choosing larger m if neces-

sary

(5.8)　　　　　　　 況(f)＝O(が)　as Z →oo　(O ＜αく1).

(15)

40 FUMIOKI ASAKURA

Next we introduce the Pleijel lemma.

Lemma 5.2. Let N（t）be a non-decreasing function defined on （0,

oo）satisfying N（t"）=O（t）as t→oo (0くα＜l ）. For α∈≡［0, ×o）we set

S(λ)＝回 浮.

Let C ＝t十iΓand Lりe a curve joining ぐand ごin c ＼[0，oo]

Then it follows

(5.9) N( ） 止 ∫プ(X)力トCt ＼S(こ ＼.

A simple proof is found in S. Agmon ［21.

We set

Jo t ―

ミp-＝Tr(A- －^)-'　　and　　 ぐ＝f十it'-’

Then （5. 9）says

況(f) 一犬

2mTr(A? －X)-'力

トCt'うTr(AT －ぐ-^.

≒Employing Proposition 5. 1, we find

(5. 10)

and

1皿(z)‾士(Tr Q'

λ)ぶ卜a

｀-ITr Q゛(ぐ

十c凶r(山州.％- ″・

Let us denote

皿 °() ＝

E且一

にQ

∫E

。(。S)1

ダ｀2;4

Qi')　((T(0) ＝9 野，μ＝μ,)

We find by （3.10 ）and （3. 11）

(5. 11)Tr ≪"≪ し゚しら3r ξ－

？＆肩丿'☆^ 絹°()

(5.12)　　呼Qi'(X)＼

On asymptotic distribution of eigenvalues for a

certain elliptic operator in a cylindrical domain

then α(x ，ξ)satisfies　　　　l

(5. 15)　　　　 ξ・Fμ2．(^，ξ)十J・Fμ2．(x，ξ 之C μ,．(x，ξ

for large z and ξ.

41

In this case, (5. 13)follows from (5. 15).

Furt

こ 余j でl こj)( ゛)sds ≒s 皿 ゜)(゙)£Cr‘ Hence

皿( か) £Cf 汽

at

(

p ‾2‾)

こ こ)we mayassurne ，ヅ; ま でノご?ey 二 万とでミe larger P)if necessary

に 几

。wing lemma

二 翼D. こ こ に

Lemma 5.3. (£. Robert (10], §7, lemma 5). Let Nl'(t)satisfy

(5. 16)

(5. 18)

and （5. 17）. Then it follows

∫:

lS－(

ム:

削げ皿O)(S)くC

絹7(Z)lOg( 子＋2)

Inserting （5. 18）into （5. 12）with X ＝t十ir, we find

(5 19) にrr Qi'(X)＼o) ・

In order to estimate the integral of Tr Qμ)，we deform the path of the

integral following T. L. Pham ［9］and D.

dl ＝

£c

dl 十

Robert[10]. For />1, we have

dl ＋

£2

心

£t

where Z-t is the segment joining Z 十le- ’ and Z 十it, £t is taken along the

circular　arc　with center O joining Z 十it BXid t －it and £3 is the segment

joining t －it and t~it^ ニ　Taking y＝d（X ）to be the variable of integration.

we find by （5.19 ）

£l＋ dx卜(::r ‾<£

‾や(log

t)m ？()'

Employing (5.19)again with d(λ)>|λ，we have

Then we obtain

(5. 21)

For j

jトCt 循(Xog

t)Ni°(t)£j　　　・　　・

しTr 邨( λ)力|£Cド ‾'(£‾’)(loμ) 皿 ≪()

0, we have

( 芦= ＼ dk十Lj £叶( dX

( 17 ）

for i>i.

42 FUMIOKI ASAKURA

where £i is the segment joining Z 十it'-' and Z 十is (e＞0) ， £, is a path in

c ＼[0, oo]joining Z 十is and Z － is and £3 is the segment joining Z － is and

Z一it^-’.We can readily see that there e χists a constant C not depending

on s such that

d丿十

On the other hand we

回

have

≦Q-^(lo μ) Ⅳ^°)().

江 口A,

ε->0　乙iJo t －λ

In this way. letting s→0, we obtain for any O＜r＜去and N

(5. 22)　 ＼N,(t)―皿'() 匯C-Xlog t) Ⅳ'() 十Q-“み‾^)゙μ

訃’煙

where C (depends on 7 and N but not on k. Summing up (5.22)with

respect to k, we find

(5. 23) N() 一八 皿O)(Z)トG-7(10 μ) 雲 皿O)(Z)

十c臨 誓 十m 一昔) -2n(-L刊巴

Substituting z for f, we obtain

Theorem 5.4. Assume the conditions ・(1.2), (1. 3), (1. 4)and (1. 5)・

Assume also (5.15).　Then it follows

(5. 24)　^(^) ＝輿 兄∫∫　　　

心 考(1 十〇(r))(2π) ん=i　"!(≫, ξ)十μβM≦i

as t→゛o for any てsuch that 0＜Tく1

Ay:e でtarry out the proof of Lemma 2.

6. We follow the proof of

Lem7

θ

ヅ プjjJ Zl とで2yoご)t ごkeibnitz formula. we have

∂g£)y・’(j'ξ)ニ心|β'十/3"

にβ　　　　　　　　

×∂fDf' ）.(x十ty, ξ力 両・

こ こ √e find thatごhave only 乱sho こ^,mi(

ヱ，ξ)心 泌，ξ)

゛he7

e

yj ツ)t)jT 回

・-

ご;t 二 で濃;じえO

く吟) ぶ1，9 ＝O for t>Co and

()o=l for t,(…)φo(t)心 励 十〇s－∫{e一.>,(

…)吟(i)心 励

＝R,(x ，ξ)十R.(μ，ξ).

(18)

謡ぶ 雪 念 謡 言t げ ぷ回 さ1 訟

(I)Estimates of R.(x ，ぐ:Since supp (p^ ⊂{ ＼y匯co} ，we find

否の(x，ξくの(x 十ty, ぐ ≦C の(ヱ，ξ)

古゛i(゙ ' ξ≦゛2卵 十ty, ξ< 洸,(゛バ

hold uniformly in t.　We denote above by ゆ(x 十ty, ξ ～Φ(エ，廻)and

碗(^ 十ty, O ～nii(x，ξ.We divide the integral as the following

Ri(x ，ξ)＝Os －∫je・

”'

44 FUMIOKI ASAKURA

＝尺2iU ，ξ 十μ^(x ，ξ 。

(n-i)　For i?2i，we have by integration by parts

R^x(x ，ξ)

づ ト ープ ダ"( －d) 匈。(y)9'>(の-'η)Pi(^ ，ξ十即)ちix 十ty, ξ 今 励・

Since ㈲ ぶご

[10]

[11]

[司

[131

[141

2 ぶ 堡 詐 ≒ 旨i 富 錨 雪 お。謡 ご

D. Robert, Comm. Partial Differential Equations, Vol.3 (1978), 755-826.

U1 こ と六 江 ご7 々:穴 昌7 に でn グ ー300.

器 言 言 昔 詣 太 太 こ ご 詐 ご ご ご に ご ‰).

（21 ）

45