Asymptotic formulae for the schrödinger operator with dirichlet and neumann boundary conditions

Vol. 55 (2005)

REPORTS ON MATHEMATICAL PHYSICS

No. 2

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR WITH DIRICHLET AND NEUMANN BOUNDARY CONDITIONS S. K A R A K I L I ~ 1 Department of Mathematics, Faculty of Art and Science, Dokuz Eylfil University, Tmaztepe Camp., Buca, 35160, Izmir, Turkey (e-mail: [email protected]) 5. ATILGAN Department of Mathematics, Faculty of Science, Izmir Institute of Technology, Gtilbah~e, Ufla, Izmir, Turkey (e-mail: sifinatilgan @iyte.edn.tr)

and

O. A. VELIEV Department of Science, Dogus University Aclbadem, Kadnk6y, 81010, Istanbul, Turkey (e-mall: oveliev @dogus.edu.tr) (Received October 6, 2003 - Revised November 19, 2004)

In this paper, we consider the Schr6dinger operators defined by the differential expression Lu = - A u + q (x)u in d-dimensional paraUelepiped F, with the Dirichlet and Neumann boundary

conditions, and obtain the asymptotic formulae for the eigenvalues of these operators. Keywords: perturbation, Dirichlet and Neumann boundary conditions, Schr6dinger operator.

Let g2 -- { E i d l mitoi; mi ~ Z, i = 1, 2 . . . . . d} be a lattice in R d with the reduced basis Wl = (al, 0 . . . . . 0), w2 = (0, a2, 0 . . . . . 0) . . . . . Wd = (0 . . . . . O, ad), { d F =- Y~i=a nifli " ni 6 Z, i = 1, 2 . . . . d} be the dual lattice of ~2, where (wi, fij) = 2 1 r S i j , (., .) is i n n e r product in R d a n d F ----- [0, a l ] × [0, a2] × . . . × [0, ad]. W e consider the d-dimensional Schr6dinger operators L D ( q ( x ) ) and L N ( q ( x ) ) , defined by the differential expression Lu = -Au

+ q(x)u

(1)

in F , with the Dirichlet boundary condition UIOF = 0

and the Neumann boundary condition 1Supported by a grant from TLIBITAK. [2211

(2)

222

s. KARAKILI(~, ~. AT1LGAN and O. A. VELIEV

Ou On [OF :

(3)

0,

respectively. Here OF denotes the boundary of the domain F, x = (xl, x2 . . . . . xd) e R d, d > 2, A is the Laplace operator in R a, and ~ denotes the differentiation along the outward normal n of OF. We denote the eigenvalues and the normalized eigenfunctions of LD(q(x)) by AN and kI-/u respectively, and of Lx(q(x)) by T N and ~U, respectively. The eigenvalues of the operators LD(O) and LN(O) are IF I2, and the corresponding eigenfunctions are u× (x) = sin 9/ix 1 sin y2x2 . . . sin Fdxd,

(4)

UV(X) : COS y l x 1 COS y Z x 2 . . . COS v d x d ,

(5)

r+ = {F 6 2r . Fi > 0, i = respectively, where in (4), V = (F 1,F 2, .. . , F d) e -71,2 . . . . ,d} and in (5), y 6 r 2+° = { F 6

7r . F i >- 0 , i =

1,2, " " , d}. r+0 It is clear that the norm of the function vr(x), F e --T- in L2(F) is

V IayI'

where A r = {or = (oq,o~2,...,o~a) e ~F • Iotil = [ y i l , / = 1,2 . . . . . d} and IA×[ is the number of vectors in A r. Thus for q(x) e L2(F) we have q(x)=

~ IA×I r ~ r+0 /x ( f ~ (q (x), v r (x)) v× (x).

(6)

Using (6) and the following obvious relations

v×(x) = v,~(x), F =

U

Fe

(q(x), v~(x)) = (q(x), v,~(x)),

ar

(q(x), vr(x)) --

and

F+O

'Cot e Ae,

1

Z (q(x), v~(x)), IA×I cteA 7

we have q(x)=

Zr+0/z(F~--) IAyl (q (x), v× (x))v× (x) re T

=

Iarl 1 #(F)IArl

~ re

=Z

r,+O

~ oteA F

1

lz( F)

(q(x), v~(x))v~(x)

(q(x), vy(x))vy(x),

So, one can write

q(x) = Z qyv×(x), rc r

(7)

223

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

1 where qy = lz(F)(q(x), vy(x)). Clearly, the formulae (6) and (7) are the same. In our study, it is convenient to use the formula (7) instead of (6) for the sake of simplicity. Without loss of generality, we can take q0 = 0. First asymptotic formulae for the eigenvalues of the Schr6dinger operator in parallelepiped with quasi-periodic boundary conditions have been obtained in papers [9-12] and the other asymptotic formulae for quasi-periodic boundary conditions in two- and three-dimensional cases have been obtained in [2, 3, 7, 8]. The asymptotic formulae for the eigenvalues of the Schr6dinger operator with periodic boundary conditions have been obtained in [4] and with Dirichlet boundary conditions in two-dimensional case in [5]. We assume that the Fourier coefficients of the potential q ( x ) satisfy the condition

Z

]qy '12(1

+ ])/t[Zm) < (30.

(8)

i F

Therefore, one can write q ,vy,(x) + O(p-P'~),

q(x) =

(9)

yIcF(pC~)

where p = m - d, F F ( p ~) = {y • ~ "0 < I×1 < p~},

1

o r < - (d + 20)'

m>

(d - 1)(d + 20)

2

+d+l

and p is a large parameter. As in the papers [12, 13] we divide the eigenvalues ly] 2 for lyl u p of the Laplace operator into two groups, where ]y[ ,-~ p means that clp < lyl < c2p and by ci, i = 1, 2 . . . . . we denote the positive independent of p constants whose exact values are inessential. For this, we let Oek = 3ko~, k = 1, 2 . . . . . d - 1, and introduce the following notation and definitions: M = Z

Iq~,l,

(10)

g ct ~ -F

Vb(P'~J)={x • R d :]lx[ 2 - I x +bl2l < pal}, El(pal'

P) =

Vb('PCq)'

U b~r(pp ~)

U (p ~1 , p) = g{d\E~(p~l, p), k

U

Y1,Y2..... vkcU(pp et)

i=1

where the intersection n~=l v×i(P '~P in E~ is taken over Yl, Y2. . . . . Yk which are linearly independent vectors and the length of Yi is not greater than the length of other vectors in F n y i R . The set U ( p '~ , p) is said to be a non-resonance domain,

224

S. KARAKILI~, ~. ATILGAN and O. A. VELIEV

and the eigenvalue I:/I 2 is called a non-resonance eigenvalue if y e U(p~l,p). The domains Vb(p'~1), for all b e F ( p p ~) are called resonance domains and the eigenvalue Izt 2 is a resonance eigenvalue if 7 e Vb(p~). As noted in [13], the domain Vb(p °'~) \ E2, called a single resonance domain, has asymptotically full measure on Vb(p~l), that is

Iz((Vb(p `~l) \ E2) [") B(p)) --~ 1, #(Vb(P ~ ) A B(p))

as p --~ c~,

where B(p) = {x e R d : Ix[ = p}, if 2 o r 2 - - 13/1 "~- (d

+ 3)or < 1

(11)

and (12)

O~2 > 20tl, 1

hold. Since oe < a--jN, the conditions (11) and (12) hold. In [1], we obtained the asymptotic formulae for the non-resonance eigenvalues of the d-dimensional Schr6dinger operators Lo(q(x)) and LN(q(x)) with the condition (8). In this paper, we obtain the asymptotic formulae for the resonance eigenvalues. The main result of this paper is that we find connection between the eigenvalues of the Schrtidinger operator corresponding to a single resonance domain and the eigenvalues of the Sturm-Liouville operators. We investigate the perturbation of the eigenvalue lYl 2, when y e Va(p(~). We assume that g = (y l, 1/2. . . . . yd) ¢ Ve~(p'~), for k = 1, 2 . . . . . d, where

el=

0..... 0 ,

e2=

0,--,0,...,0

.....

ed=

0 . . . . ,0,

.

a2

This relation implies that iv k] > ~p~l,

¥k = 1, 2 . . . . . d.

(13)

The case ~ = el, i = 1, 2 . . . . . d, was considered in [6] where a different but simpler method was used and better formulae were obtained. Since there is no intersection between two methods of investigations of the cases /t - - e i and ~ ~ el, i = 1, 2 . . . . . d, we study them in different papers. It is obvious that for y ~ Vek(pal), for k = 1, 2 . . . . . d, the eigenfunctions uz(x) and vy(x) have the forms (4) and (5), respectively, where yk satisfies Eq. (13). Clearly, 1

Vz(X ) --

%

e i(°~'x),

IA×I c~cAy where A z = {oe = (Otl,Ot2. . . . . o~,t) e I~a : Ioeil = lyil,i = 1,2 . . . . . d} and IA×I is the number of vectors in A×. Using these, it is not difficult to verify that for all

225

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

r satisfying (13) and for all a

1 IAy~( 1 Va(X)V×(x)--[Aa--I

e

F(p~), the following relations hold:

ei(×',x))( B ei(a'x})

B

?"EAa

o:tAy

1 1 ~eA A~ ei(a'x}-- 1 ~ -IAal I-4xl Y a~6 y_yt 1,4al y 6Aa vx-x"

(14)

since IAxl = lAx_ x, I = 2a for every y satisfying (13) and for every y' e F(p~), !

because all components of ~, and y - y are different from zero. The set of the vectors a 1, a 2 , . . . , a s, where s = IAal and clearly

A~I =

Aa2

Val

Aa s = Aa,

. . . . .

=

Hence in (14), the vector a can be replaced by a 1, a s equalities and using (15), we get

/)a 2 . . . . . 2 .....

Aa consists

VaS= Va.

as. Summing

(15)

the obtained

s

=Z

Therefore,

Z

yttA a

k=l

B

yttAa

vttAa

B qy, vy,(x)vy(x)= ~_~ qy, v×_×,(x), }/tEAa

(16)

y eAa !

since q×, = q~ for all y' e Aa (see (15)). Clearly, there exist vectors al, a2, .. . , an e _rz such that n

aaj f-) aak

F(P~) = g a a j ,

=

0,

Yj ¢ k.

(17)

j=l

In (16), replacing a by and using (17), we get

Z

aj

for j = 1, 2 . . . . . n, summing all obtained n equalities

q×'vx'(x)vx(x)=~ Z

q×,v/(x)v×(x)

j = l y 'eAaj

yl6F(p~ )

= ~-]~ Z j=l

q×,v~,_~,,(x)= ~

}"r eAaj

q×tv×_×,(x).

(18)

~/tP(p c~)

In the same way, we have

Z y'eF(p~)

qy'v×'(x)ur(x)= Z

q×'ue-e'(x)"

(19)

y'eF(p~)

To obtain the asymptotic formulae for the eigenvalues of the operators L D ( q ( x ) ) and LN(q(x)) we use the following well-known formulae:

226

S. KARAKILI~, $. ATILGAN and O. A. VELIEV

(_AN -- Iyl2)(qJU, ur(x)) = (qdN, q(X)U~,(X)), (TN -- lTI2)(qbN, vr(x)) = (~U, q(X)V~,(X)),

(20)

where (., .) is the inner product in Lz(F). Substituting the decomposition (9) of q(x) into both formulae (20), we get

Z

(A N - - 1719-)(~N, u×(x)) =

q?/(qdN, Yy, Uy(X)) -ff O(p-P°t),

y'cI'(p~) ( T N -- IyI2)((I)N, 1)y(x)) =

qy,(dPN, V ,Vy(X)) -k- O(p-P°~).

Z yI~F(pC~)

Hence by (18) and (19), we obtain (AN

--lTI2)b(N, Y)--

Z

q,/b(N, 7 - y') +

O(p-P'~),

y~eF(p c~)

(TN -- lYl2)c(N, y) =

B q×,c(N, 7 - 7') + O(p-P'~), yter(pC~)

(21)

where we use the notation b(N, y ) = (kilN, big) and c(N, y ) = (dPN, IJg). Also (20) and (9) imply that

b(N, ~) = (qdN' q(x)u~)

=

AN --[~[2

c(N, p') =

(¢PN, q ( x ) w ) TN - - ] ~ l 2 =

Z

qY'

b(N, ~ - g')

gfEP(pU) Z

qy' c(N, _~ - Y') + TN

ytEP(p e~)

IP'I 2

r satisfying the conditions [AN--]~I2[ for every vector p" C 3' 1 ~1 ~r-

+ O(P-P°~)'

AN - - [ ~ ] 2

>

O(p-P'~),

1/0eel

(22)

and [TN--[~'I2[ >

•

Let ITI 2 be a resonance eigenvalue, of o r d e r p2, of the operators LD(O) and LN(O), i.e. g 6 (Nik_~ Vyi (p=k)) \ Ek+x, k > 1, Yi ~= e j, i = 1, 2 . . . . . k, j = 1, 2 . . . . . d, [Yl "~ P. Define the following sets: ~k - , Bk(Yl, Y2. . . . , ~/k) = { b • b = ~....~rliYi, l"li C Z, Ibl < 12p }21"k+I i=1

Bk(g) = g + Bk = {y + b : b 6 Bk},

Bk(g, Pl) = Bk(g) + P(plP=), where Pl is the integer part of @ ,

and the matrix

C(y, y1 . . . . . Yk) = (cij) =

qhi-h j, Ihil 2 ,

i 7/=j, i=j,

227

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

for i, j = 1, 2 . . . . . bk, where by hi w e denote the vectors of Bk(y, the number of vectors in Bk(y, Pl). Writing Eqs. (21) for all hi c Bk(y, Pl), one has (AN -- [hil2)b(N, hi) =

Z

q×,b(N, hi - y') + O(p-P~),

Pl)

and bk is

i = 1, 2 . . . . . b~,

y/c[,(pC~)

(TN -- [hi[2)c(N, hi) =

E q×,c(N, hi - y') + O(p-P'~), y/EI"(por)

i = 1,2 . . . . . bk. (23)

The similar system of equations for quasi-periodic boundary condition was investigated in [12] and [13]. Arguing as in Lemma 6 of [12] and in Theorem 1-(c) of [13], we can d3da easily obtain that bk = O(p7 ), b(N, h i - y ' ) = O(p-pC~), c(N, h i - y ' ) = O(p-P~), for hi - y' ¢ Bk(y, Pl), if o~+1 > 2(C~k+ (k - Dot) and we get the following system of algebraic equations:

(C - A N I ) [ b ( N , hi) . . . . . b(N, hbk)] = [O(P -p~) . . . . .

O(P-PC~)],

(C - T N I ) [ c ( N , hi) . . . . . c(N, hbk)] = [O(P -p~) . . . . . O(P-P~)],

(24)

for all eigenvalues AN and TN satisfying 1

lag-lY121 < ~p

2~

1,

ITN--lY121 <

p2~.

(25)

k THEOREM 1. Let y 6 (["]/=1 V×i(P~k)) \ E~+I, k = 1 , 2 , . . . , d 1, Yi (= ej, i = 1,2 . . . . . k, j = 1,2 . . . . . d, Iv[ ~ P . Then, for every eigenvalue ~i(Y) of the matrix C(y, Y1. . . . . Yk) satisfying [)~i- ly121 < 3p.1, there is an eigenvalue AN of the operator LD(q) and an eigenvalue TN of the operator LN(q) such that d d d--I AN = ,ki(y) + O(p -(p-~3 )o~+~-), TN = ~'i(~) -~ O(P-(P-~3a)c~+@).

Proof: Due to general perturbation theory, there a r e A N and Z N satisfying (25). Therefore, we can use the systems (24). Let )~i be any eigenvalue of the matrix C and vi = (vii, vi2 . . . . . ribs) be the corresponding normalized eigenvector ([[vi[[ = 1). Multiplying Eqs. (24) by vi, we have ((C - A N I ) [ b ( N , hi) . . . . . b(N, hbk)], l)i) = ([0(t O-p°~). . . . . O(p-Pa)], Vi),

((C - T N I ) [ c ( N , hi) . . . . . c(N, hbk)], vi) = ([O(p -p~) . . . . . O(p-P'~)], vi).

(26)

Since C is a symmetric matrix and Cvi = ,kivi, the right-hand side of (26) has t h e form bk ((AN -- ~.i)Vi, [b(N, hi) . . . . . b(N, h b k ) ] ) = (AN -- )~i)( Z PijRhj (x)' kI'tN) ' j=l z

((TN

--

bk

\

)~i)Vi, [c(N, hi) . . . . . c(N, hb~)]) = (TN -- )~i)( __Y~VijVhi (X), CbN). \

j=l

228

S. KARAK1LII~, ~. ATILGAN and O. A. VELIEV

Therefore using I([O(p -p'~) . . . . .

O(P-Pa)],

vi)l

=

O(P -(p-d3d)cz) and (26) we get

bk = O(p--(p--d3d)a), (AN -- )',.i)( E UijUhj (X), kIIN) j=l

bk = O(p--(p-d3d)°t). (TN -- ~i)( E I)ijUhj (X), f~)N)

(27)

j=l

So, we need to prove that there exists N such that

UijUhj(X), kIIN

> C3p

2 ,

VijUhj(X), ¢~)N

j=l

> C3t0 2 ,

(28)

j=l

from which the theorem follows. For this, we first consider the decomposition of the matrix C(F, }'1 . . . . . }'k) as C = A + B, where the matrix A is defined as A = ( a i j ) , aij = 0 if i ~= j, ajj = [hj 12 , for i, j = 1, 2 . . . . . bk. Here [hi 12 is an eigenvalue of the diagonal matrix A and ej = ( 0 , 0 . . . . . 1. . . . 0) is the corresponding eigenvector. We denote by l)i(hj) ~" (Yi, ej) -~- Yij, the j - t h component of the vector vi. Note that B -----C - A is a symmetric operator for which the diagonal elements are zero, and the sum of the elements in each row of B is less than M - ~-~×e~ Iqy]. Therefore, the eigenvalues of B are smaller than M and [[Bll < M. Now, multiplying the equation Cvi = ,~ivi by ej and using that A and B are symmetric matrices, we obtain

(Xi --]hjl2)vi(hj) = (vi, Be j).

(29)

Using Eq. (29) and the Bessel inequality, we have

E

j:lLi-lhjl2l> l p°q

Ivi(hj)12 ~-

E

j:[zi-lhjl2l>_l pal

I(Bvi, ej)l 2 _ O(p_2=i)" IZi _ Ihjl212

(30)

Now, we prove the first inequality in (28). By the Pythagorean relation, we have

bk

bk

j=l

N=I

=

2

j=l

E

(

N:lAN-lYl21~lp2al

bk j=l

bk N:IAN_IF}2I>l p2al

j=l

2.

229

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

First, we consider the second sum in the last expression, that is

bk (Evi(hj)Uhj(X)'*N)

E N:lAN_lyl2l>l p2Cq j=l

=

E

(

2 Vi (hj)Uhj (X), tlIN)

E

N:[AN-ly[21>l p2Cq j:lLi-lhjl2l
+(J:l~-i-Ihjz121>_lp ~1 vi(hj)Uhj (X), kilN) 2 <2

E

(

Vi (hj)Uhj (X), kilN) 2

E

N:IAN-IyI2I>I p2al j:lAi-lhjl2l
+2

E

(

N:IAN-IYl21> lp2al

E vi(hj)uhJ (X)' rIJN) 2. j:lLi-lh jl21>l pcq

(32)

3t~l 1,~"1 and the condition of the theorem I)~i-[yl21 < g~. Now, notice that I~.i-lhjl21 < ~,~ together imply IlYl 2 - Ihjl21 < ½p=l Therefore, using Eq. (20) and the well-known formula

1

1

--

AN - - I h j l 2

-]-"'-[- (Ihj[ 2 - l Y l 2 ) k-1

-k O ( p -(k+l)~l)

(33)

(AN --Iy[2) k

AN --lYl 2

for 1 2a

IAN --19/121 > ~P

and

'

ol

ityl = _ ihj]21 < ~p 1,

(34)

we have

E

l)i(hj)(Uhj (x)' kilN) 2

E

N:IAN-1~I2I>l p2eq j:l)~i-lhjl21
E

E

N:IAN_Iy[2I> lp2Cq j:]Li-lhj 121< lp '~1

< ( k + 1)

E

2

AN--~ ~jl 2 F(j)

2

E

N:IAN_lyl21>lp2Cq j:lLi-Ihj 121< lpCq + ( k + 1)

F(j) (AN 7 ~12)2 (WI2 E E N:IAN--IYI2[>½,o2Cq j:lXi-lhjl2l
Ihjl 2) 2

+""

S. KARAKILI~, ~. ATILGAN and O. A. VELIEV

230

F(j)

+(k+ 1)

(AN - - l y l 2 ) k

Z

O(p_(k+l)~,)F(j)2,

N:]AN--]yI2[> l p2al j:lXi-thjl2l< l peZl

-t- (k + 1)

~

([Yl 2 _ [hj[2) k-1 2

Z~,

(35)

N:lAN--lYl21> l p2eq j:l,ki-lhjl2l
F(j) = (vi (hj)q (x)uhj (x), ~N). To calculate the order of each term in (35), we use (34), the Bessel inequality with respect to the basis {qJN}N~__],and the orthogonality of the eigenfunctions Uhj(X). Thus, we obtain

(ANFZ(~I2)s([~I 2 Z .j:]Li-lhj 121
N:IAN_IYI21>/p2cq

(1)i (hj)q(X)Uhj (x), k~tN)

= N:IAN--[yIZ]> Z ½p2al

Z j:l)~i-]hj[2l< l p°q

Z

(q(x)

N:lAg-lYl2]> l p 2 ~ l

C5/O-4s°q

~ CSp -4sal

[h j[2) s-1 2

(lY[ 2 _

~

vi(hj)uhj(X)([y[ 2

[hj[2) s-1

j:l,~i-lhjl2t< l pCq

Z N:IA N _[yt21>lp2cq

(q(x)

Ihjl2) s-1 2

(i N _ lYl2y

.N

)2

' (AN--Iy[2) s

(q(x)

Z ui(hj)Ighi(X)(lY[2-- ]hjl2)s-l' kilN) 2 J:[)~i-Ihj 12]
vi(hj)uhj(X)(l~gl2- [hj[2)s-1) 2

~ j:lZi-lhj 12l< lpal

c5p-4s'~lM2

~

Ivi(hj)12llyl2 --Ihj]212(s-l)

j:lZi-[hj ]21
< c5p-4S°qM2p2(S-1)'~l

~

Ivi(hj)l 2

J:[)~i-lhj 12]< lpCeI = O(p-2(s+l)~l ) for s = 0 , 1

. . . . . k.

Now, let N1 be the number of of the last term in (35) is

hj

satisfying

I,ki -Ihjl21 < ½p,~l. Then,

the order

231

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

N:IAN--1~I21> lp2~l

O (/9 -(k+l)~l )(U i (hj)q(x)uh,i (x), IJIIN) 2 E j:l)~i-lh,il21
] O (p-(k+l)cq)12l (vi (h .)12l(q (x)uhj (x), kilN)12 N:IANTIVI21 > lp2~l

<--C6Nlp-2(k+I)°q

j:l;vi-lhj 121
Z I(vi(h,i)l 2 (q(x)uhj (x), j:lLi -[hj 121< l p ~ N:IAN-IYI2I>½p2~I

<--C6Nlp-2(k+l)°q

Z j:[Zi -[hj 12[
d

O(pT°~d), and

~N)[ 2

IIq (x)uhj (x)II2

we can always choose k in

O(p -2(k+l)°q) such

that

N 20 (p-Z(k+l)cq) = O (p-ZCq). Also, using Bessel's inequality, the orthogonality of the eigenfunctions ur(x ) and (30), we have

Z N:IAN--lYl21 > 21-p2eel -<

(

Z

l)i(hj)bthJ(x)' ddN)

2

j:iZi-]hj 12[R lpeq

Z

vi(hj)uh i (x) 2

j:[Zi -[hj 12t>.l poq

=

y~

Ivi(hj)l 2

j:lZi -1hi 121> lpCq = O (p-2Cq).

(36)

Therefore, from (31) it is clear that

Z ( k v i ( h j ) U h J (x)' q/N) 2 N:IAN_IyI2[<_lp2eq j=l

= C4 -- O ( p - 2 ~ l ) .

(37)

1 2cq Now, taking into account that the number of indices N satisfying [AN- ly121 < ~p is smaller than c7p d-l, we get the first inequality in (28),

bk t ~--~ Vi (hj)blh i (X), I-IJN t > c8p d-1 2 j=l

Similarly, we can prove the second inequality in (28). So, dividing both sides of

232

S. KARAKILI(~, 5}. ATILGAN and O. A. VELIEV

the first and second equations in (27) by

(~VijUhj(X),~N)

(~VijVhj(X),~N),

and

j=l

j=l

respectively, using the inequalities in (28), we obtain the formulae in the theorem. [] Now, we investigate in detail the eigenvalues of LD(q(x)) and LN(q(x)) in a single resonance domain V~(p~), where 3 7~ ek for k = 1, 2 . . . . . d. Namely, we find the relation between the eigenvalues of LD(q(x)) and LN(q(x)) in a single resonance domain and the eigenvalues of the Sturm-Liouville operator. In order that inequalities (11) and (12) be satisfied, we can choose o~, O~1 and c~2 as follows 1

P2

~--d+p

d+p

2p2 + 1 Ol2 -- - - ,

d+p

where

and [ @ ] is the integer part of the number p-S 3 " Let g E V~(p ~1) \ E2, where ~ ¢- eg for k = 1,2 . . . . . d, 6 6 gF and ~ is minimal in its direction. Consider the following sets

BI(3) =

{

11

/

b " b = n6, n 6 Z, Ibl < ~ p ~c~2 ,

B I ( y ) ---- y + B l ( 3 ) = {y + b " b c Bl(3)},

BI(y, P I )

= B I ( y ) + F(plPC~).

As before, denote by hi, i = 1, 2 . . . . . bl the vectors of BI(F, Pl), where bl is the number of vectors in BI(Y, Pl). Then the matrix C(V, 6) = (cij) is defined as ( c i j ) ~- q h i - h j , if i 7~ j, (cii) = [hil 2, where i , j = 1,2 . . . . . bl. Also we define the matrix D(F, 3) = (cij) for i, j = 1, 2 . . . . . a b where hi, h2 . . . . . hal are the vectors of BI(I', pl) ["]{g + n6 • n c Z}, and al is the number of vectors in BI(F, Pl) A{F + n3 • n ~ Z}. Clearly al = O(pza2). Now we prove 1 ilhjl2 _ ihil21 > ~ p 2,

Vi < al,

Vj > al.

(38)

1 t~ 1~2 By definition, if i _< al then hi --- V + n3, where In31 < ~t~ 2 -t- plp ~. If j > al 1 ~lc~ 2 then h j = F -t- S(~ + a, where Is6[ < ~ r . 2 , a e F ( p l p ~) \ 3R. Therefore

Ihjl 2 - Ihi] 2 = 2(y, a) + 2s(~, a) + 2s(F, 6} + Is312 + [a[ 2 - 2n(y, 3) - In~l 2. Since g ~ Va(P~Z), la[ < pip ~, we have 12(g, a)l > f )ct2 - C9D 2et.

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

233

The relation ~, c V3(p ~ ) and the inequalities for s and n imply that 2s(y, 8) + 2s(y, a) + lal 2 - 2n(v, 8) = o(pl~2q-°~l), 1

iis~l 2 _ In8121 < ~ p 2 + clop½~2+=. Thus (38) follows from these relations, since ~ot2 1 + oil < ~ 2 a n d ~Ot 1 2 q-Or < O~2. The eigenvalues of D(V, 8) and C(y, 8) lay in M-neighbourhood of the numbers Ihkl 2 for k = 1, 2 . . . . . al and for k = 1, 2 . . . . . bl, respectively. The inequality (38) shows that one can enumerate the eigenvalues ~'i (i = 1, 2 . . . . . bl) of C(V, 8) such t h a t )~i for i < al lay in M-neighbourhood of the numbers Ihkl 2 for k _< al and ~.i for i > al lay in M-neighbourhood of the n u m b e r s Ihkl 2 for k > a l . Then by (38), we get 1 c~2 I ~ , i - Ihjl21 > ~ p ,

for i < a l , j > a l

and i > a l ,

(39)

j
THEOREM 2. Let y c V~(p ~1) \ E2 where 3 # ek for k = 1, 2 , . . . , d. For any eigenvalue )~j of the matrix D(y, 8) there exists an eigenvalue )~i(j) of the matrix C(y, 8) such that 1 )~i(j) = "~j "~- O(p-2C~2) •

Proof: Let ~i be an eigenvalue of C(V, a), and vi = (vii, vi2 . . . . . Viel) be the corresponding normalized eigenvector. Denote by ej = (0, 0 . . . . . 1, 0 . . . . . 0) the eigenvector of A(V, 8) corresponding to the j-th eigenvalue ]hjl 2, w h e r e ¢

A(~,, 8) = (aij) = {

O,

i # J

Ihi 12,

i = j,

for i, j = 1, 2 . . . . . bl. Let B = C(y, 8) - A(V, 8). Clearly, we have

()~i - Ihjl2)vij = (ui, Be j).

(40)

Substituting the orthogonal decomposition Bej = Y~I (Be j, ek~ )ek~ in (40), we obtain al

bl

(&i - Ihjl2)vij = ~'-~(Bej, ekl)Vik 1 qkl=l

Z

(Bej, ek~)Vikl.

(41)

qhkl-h~Vikl.

(42)

kl=al+l

It is clear that (Be j, ekl) = qhkl_hj, and thus al

(Xi -Ihjl2)vij - ~_~ qh~,-hjVi~, = kl=l

bl

~ ki=al+1

For i = 1, 2 . . . . . al, using (39) and (40), we find the following estimation for the

234

s. KARAK/LI~, ~. ATILGAN and O. A. VELIEV

right-hand side of Eq. (42)

qhk, -h J ~i(Vi,-- Bekj) ~-~k~,-2

~.al + qhkl -h i Vik, l = I ~ kl=

1

kl=al+l

bl <

11Vi II IIB II][ekj [[

S

[qhlq-hJ

kl=al+l

< lp-~2M2

[)~i_[h/q[2]

(43)

= O ( p -°~2)

for all j = 1, 2 . . . . . al. Now, writing Eq. (42) for all h j, j = 1, 2 . . . . . al and using (43), we obtain the system

( D ( y , 8) - )~iI)[Vil , vi2, . . . , vial] =-

[O(/9-a2),

O(I O-a2) . . . . .

O(p-a2)].

(44)

Define the matrix D'(y, 8) = (di~) as d~i = [hil 2 for i = 1, 2 . . . . . bl, di~j = qh j-hi for i # j, i, j = 1, 2 . . . . . al and di~ = 0 for i > al or j > al. Then the spectrum of the matrix D'(y, 8) is spec(D') = s p e c ( D ) U { I h a l + l ] 2 , . . . ,

[hbl [2} _ {~1 . . . . . ~al, ]h~1+1[2 . . . . . Ihbl [2}.

Let us denote by wj = (w jl . . . . . Wjal, 0 . . . . . 0) the normalized eigenvector corresponding to the j-th eigenvalue of the matrix D', when j = 1, 2 . . . . . al and by Wk = (0, 0 . . . . . . 1, 0 . . . . 0) the eigenvector corresponding to the k-th eigenvalue of D', when k = a l + 1, al + 2 . . . . . bl. Now, using the system (44), we obtain

(D' - )~iI)[Vil , 19i2. . . . .

Pibl]

)~iI)[Vil, vi2 . . . . .

:

[(D -

:

[O(p-C~2) .....

Vial],

( [ h a l + l 12 -

O(p-e~2), (ihal+ll 2 _

)~i)Vial+l . . . . .

~i)Uial+ 1 . . . . .

(]hb I 12 -- )~i)Vib 1]

(]'hb 1 [2 __

~,i)l)ibl]"

Taking the inner product of the last system by w j, j = 1, 2 . . . . . al, using that D r is symmetric and Dtwj = ~jWj, we get a1

(~i - ~j) Z

al

vikWjk = E

k=l

O(P-~Z)wjk'

(45)

k=l

where, by the Cauchy-Schwarz inequality,

k=l

k=l

k=l

l

Since al = 0(p7~2), we have al

()~i - "~j) E k=l

3

VikWjk = 0(p-4'~2).

(46)

235

ASYMPTOTIC FORMULAE FOR THE SCHRI)DINGER OPERATOR

So we need to show that for any j = 1, 2, . . . , al there exists Vi(j) ~

Vi,

such that

al

(47)

y 2 UiktOjk = [(/Ji, tOj)] > C l l P - I ~2. k=l bl w/= Zi=l(Wj, vi)l) i

For this, we consider the decomposition bl

al

bl

1 = ~ I/wj, ~,/I2 = ~ I(~, vile + ~ i=1

and Parseval's identity

IImj, ~i/I2

i=a I +1

i=1

First, let us show that bl

Z

](wj, vi)]2 =

3

O(p-7c~2).

(48)

i=al+l al Using that wj = Y~k=l(Wj, ek)ek, Eqs. (39) and (40) and Bessel's inequality, we obtain the estimation: bl

Z

bl

[(w/'vi)12= ~

i=al-rl

i=al+l

al

2

(~Wjkek, vi) k=l

bl

al

i=al +1

=

i=al+l

bl

<

~

2

k=l

al

2

k=l

bl

~

( B e k , IJi)

p - 2 - 2 ( Z l W j k l l ( B e k , Vi)[)

i=al+l

_<

bl ~

2

al

~-2~2ia11~ IwJki2iISe~,viii 2

i=a 1+1

k=l al

aI

k=l

k=l

al

<

[amlp-2°~2~ Iwj~1211nekll2 k=l al

<--

MZlallp-2c~2Z IWjkl2=

3

O(/9-2c~2)"

k=l 3

Therefore, y2ia!l I(tlOj, Vi)I 2 = 1 - O(p-7 ~2) from which follows that there exists vi(j) such that (47) holds. Dividing both sides of (46) by (47), we get the result. []

236

s. KARAKILI~, ~. ATILGAN and O. A. VELIEV

Now, using the notation hi = / y - (~)3

! y+(k~.!)3

if i is even, if i is odd,

for i = 1, 2 . . . . . al (without loss of generality assume that al is even) and using the orthogonal decomposition for g c r , × = fl + q + v(/3))8, where ficH~--{x~IR

d'(x,6)=0},

IcZ,

vc[0,1)

we can write the matrix D(?/, 3) as O(g, 8) = [fi[21 q_ E ( y , ,~),

where E(V, 8) is: (l -t- 11)21312

q3

q-~

q_s

(l -- 1 + V)21312

q-2s

q~

q-2~

(l + 1 + I))21312

E(y, 3) ----

l

Denote

k n~=

--

al ~- -1-V)21312

if k is even,

k21 2

if k is odd.

The system {e i(nk+v)s : k = 1, 2 , . . . } is a basis in L2[0, 2re]. Let T ( y , 8) -- T ( Q ( s ) , fi) be the operator in ~2 corresponding to the Sturm-Liouville operator T, generated by -lSI2y"(s) + Q ( s ) y ( s ) = lzy(s), y(s + 270 = ei2~v(~)y(s),

(49)

where oo

Q(s) --

,2--,qnk~ , k=l

einks,

qnk~ =

( q(x),

Z

)

ei(~'x) '

s = (x, 8).

~EAnk 8

it means that T ( y , 8) is the "infinite matrix (Te i(l+nk+v)s, ei(l+nm+v)s), k, m = 1, 2 . . . .

ASYMPTOTIC FORMULAE FOR THE SCHRI3DINGER OPERATOR

237

THEOREM 3. Let y c V~(pal) \ E2 where 8 ~ ek for k = 1, 2 . . . . . d. For every eigenvalue tzi of the Sturm-Liouville operator T(y, 8), there exists an eigenvalue ~ of the matrix E(y, 8) such that 3

/Zi = /~i -]- O ( p - 7 ~ 2 ) ,

where i = 1, 2 . . . . . al. Proof.: Decompose the infinite matrix T(V, 8) as T(V, 8 ) = A + B, where the matrix A = ('aij) is defined by a"~j = 0 if i # j, "aii ~--- I ( l - i

if i is even,

V)812

I(l + !@ + v)812

if i is odd,

for i , j = 1,2 . . . . and B = T(y, 8 ) - A . Let/xi be an eigenvalue of T(y, 8), and ui = (Uil, ui2, ui3 . . . . ) be the corresponding normalized eigenvector, that is Tui = [.ziRi. Denote by ej = (0, 0 . . . . . 1, 0 . . . . ) the j-th eigenvector of the matrix A. Then, the corresponding j-th eigenvalue of A is I(J'+v)SI 2, that is Aej = I(J'+v)sleej, where j' = l - ~ if j is even, j' = l + @ if j is odd, for j = 1, 2 . . . . . One can easily verify that (/zi - [(j' + v)Sl2)uij = (ui, Bej),

(50)

and using the orthogonal decomposition Bej = )--~=l(Bej, ek}ek, we get (]Ai --

[(j'

+ V)SI2)uij = ~ ( n e j , k=l

ek)Uik,

and since (Be j, ek) = q(nk-nj)~, al

(#i - I(J' + v)612)uij

oo

~

q(nk--nJ )3uik ~- Z

k=l

q(nk--nj)aUik.

(51)

k=al+l

Now take any eigenvalue /zi of T(y, 8), satisfying I / z i - I(i'-t-v)SI2l < suplQ(s)l for i = 1, 2 . . . . . a l, where

+

1

if/iseven

2

if i is odd.

The relations ~/ ~ V~(p ~1) and 7 = fl + (l + v)6, (fl, 8) = 0 imply 12(y, 8) + 18121 = I(l + o)1812 + 18121 < p~l, Therefore using the definition of i' and j', we have

la181 I(i' + v)SI < - + ca3p ~1, 4

III < c12p ~1.

238

S. KARAKILI~, ~. ATILGAN and O. A. VELIEV al

for i = 1, 2 , . . . - z 2

for j > a l .

and la16[ I(J' + v)61 > - - c14,o c~l , 2 o~2

Since [ a 1 [ > c 1 5 p ~- and a z > 2 O q ,

we have (52)

1](i' + v)~l a - I ( f + v)~lZ[ > Cl6P c~2, for i < 5~, j > al, which implies [/z~ - I(J'+ v),~121 =[I/z~ - I(i' + v)SI21 - II(J'+ v)~121 - I(i'+ v)~12ll

(53)

> Cl7P c~2,

for i = 1 , 2 Thus,

.... 9, j>al.

oo

oo

Z

q(nk-'~9 '~uik =

k=al4-1

Z oo

_< ~

(ui, Bek)

Iq(,,k-,~j)~[

k=alq-1

/-Li Z ~ - £ ~ ~)~l 2

iq(,,~_~,i)e I

k=al+l

c,o

IluillllBIllle~ll
(54)

C18/? -~2 ,

since I[B[[ _< M. Indeed, B corresponds to the operator Q : y --+ Q(s)y in L2[0, 27r], which has norm sup [Q(s)[ _< M. Therefore writing Eq. (51) for all j = 1, 2 . . . . . al, and using (54) we get the system ( E ( y , (~) - ~ i I ) [ U i l , ui2 . . . . .

uial] = [ O ( p - c ~ 2 ) , O ( p -0~2) . . . . .

O(p-c~2)].

Using ui = Y~j~I uijej, Eqs. (50) and (53), we have

~_~

luejl 2 = j=al+l

and thus

(ui, Be j) _ ~;~)Si2

2

= 0(p-2~2)

j = a l +1 ]J~i al

luijl 2 = 1 - O(p-2c~2). j=l

Taking the norm of the vector (see (55)) [Uil , Ui2 . . . . .

blial] -= ( E ( y ,

6) -- I Z i I ) - l [ O ( p

-c~2) . . . . .

O(p-C~2)],

we get V / l - - O ( P -2°~2) = II ( E ( v ,

6)

- [ z i I ) -1 II o(~d-~p -~2)

or r a i n [~i -- ~ j [ =

J

O (~r'~/9-c~2)

v/1 - o(p-2.~)

3 m_ O ( p - ~e~2),

(55)

ASYMPTOTIC FORMULAE FOR THE SCHRODINGER OPERATOR

where the minimum is taken over all eigenvalues the result follows.

239

~ j of the matrix E(V, 3). Thus, []

Let g ~ V ~ ( P " I ) \ E 2 where 3 7~ek for k = 1,2 . . . . , d and H~ = {x 6 R a • (x, 3 ) = 0}, thus we have the following main result: THEOREM 4. For every fl ~ H~ , Ifll ~ p and for every eigenvalue Izi(v(fl)) of the Sturm-Liouville operator T(y, ~), there is an eigenvalue AN of the operator L o ( q ( x ) ) and an eigenvalue T x o f the operator L u ( q ( x ) ) satisfying 1

AN = 113]2 -t-/zi + O(/0--2c~2), 1 TN = ]fl12 ._~ ]z i 71_ O ( / o - ~ 2 ) . N Proof: From Theorem 3 and the definition of E ( y , 3), there exists an eigenvalue 3.j(i) of the matrix D(V, 3), where y has a decomposition y = fl + (j + v(~))& satisfying ~j(i) = [/~12+#i + 0 ( p - 3 ~ 2 ) . Therefore, the result follows from Theorem 2 and Theorem 1. [] REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

~. Atdgan, S. Karakah~ mad O. A. Veliev: Turk. J. Math. 26, 215 (2002). J. Feldman, H. Knorrer and E. Trubowitz: Invent. Math. 100 (1990), 259. J. Feldman, H. Knorrer and E. Trubowitz: Comment. Math. Helvetica 66, (1990), 557. L. Friedlanger: Commun. Partial Differential Equations 15 (1990), 1631. O. H. Hald and J. R. McLaughlin: Memoirs of AMS. 119 (1996). S. Karakahq, O. A. K Veliev and ~. Atdgan: submitted to Turk. J. Math. (2004). Yu. E. Karpeshina: Math. USSR-Sb 71 (1992), 701. Yu. E. Karpeshina: Commun. Analysis and Geometry 3 (1996), 339. O. A. Veliev: Dokl. Akad. Nauk SSSR 268 (1983), 1289. O. A. Veliev and S. A. Molchanov: Functsional Anal. i Prilozhen 19 (1985), 86. O. A. Veliev: Functsional Anal. i Prilozhen 21 (1987), 1. O. A. Veliev: Teor Funktsional Anal. i Prilozhen 49 (1988), 17. O. A. Veliev: The Periodic Multidimensional Schrodinger Operator, Part 1, Asymptotic Formulae for Eigenvalues. University of Texas, Mathematics Department, Mathematical Physics Preprint Archive, 01446.Dec.4. (2001).