Efimov’s effect for partial integral operators of Fredholm type Текст научной статьи по специальности «Математика»

EFIMOV'S EFFECT FOR PARTIAL INTEGRAL OPERATORS OF FREDHOLM TYPE

Yu. Kh. Eshkabilov, R. R. Kucharov

Department of Mechanics and Mathematics, National University of Uzbekistan,

Tashkent, Uzbekistan yusup62@mail.ru, ramz3364647@yahoo.com

PACS 02.30.Tb

We study the existence of an infinite number of eigenvalues (the existence of Efimov's effect) for a self-adjoint partial integral operators. We prove a theorem on the necessary and sufficient conditions for the existence of Efimov's effect for the Fredholm type partial integral operators.

Keywords: essential spectrum, discrete spectrum, Efimov's effect, partial integral operator.. 1. Introduction

Linear equations and operators involving partial integrals appear in elasticity theory [1-3], continuum mechanics [2,4-6], aerodynamics [7] and in PDE theory [8,9]. Self-adjoint partial integral operators arise in the theory of Schrodinger operators [10-13]. Spectral properties of a discrete Schrodinger operator H are tightly connected (see [13,14]) with those of the partial integral operators which participate in the presentation of operator

Let Q and Q2 be closed boundary subsets in RV1 and RV2, respectively. The partial integral operator (PIO) of Fredholm type in the space Lp(Q1 x Q2), p > 1 is an operator of the following form [15]

Here k0,ki,k2 and k are given measurable functions on Qi x Q2, Q2 x Qi x Q2 and (Qi x Q2)2, respectively. All integrals have to be understood in the Lebesgue sense.

In 1975, Likhtarnikov and Vitova [16] studied spectral properties of partial integral operators. In [16], the following restrictions were imposed: ki (x,s) G L2(Qi xQi),k2 (y,t) G L2(Q2 x Q2) and T0 = K = 0. In [17] spectral properties of PIO with positive kernels were studied (under the restriction T0 = K = 0). Kalitvin and Zabrejko [18] studied the spectral properties of PIO in the space Lp,p > 1. Kernels of PIO in all mentioned articles are functions of two variables and T0 = 0. In [19], a full spectral description of self-adjoint

H.

T = To + Ti + T2 + K, where operators T0, T1, T2 and K are defined by the following formulas:

(1)

Tof (x,y) = ko(x,y)f {x,y),

PIO in the space C ([a,b] x [c, d]) with continuous kernels was given. Self-adjoint PIO with To = 0 were first studied in [10], where theorem about essential spectrum was proved. The finiteness and infiniteness of a discrete spectrum of self-adjoint PIO arising in the theory of Schrodinger operators was investigated in [11-13].

In [20], PlOs in some functional spaces were investigated and a number of applications were considered. Some important spectral properties of PIO in the space L2 are still open. The present paper is dedicated to the mentioned problem.

We study the existence of an infinite number of eigenvalues (the existence Efimov's effect) for a self-adjoint partial integral operators.

2. The notations and necessary information

Let A be a linear self-adjoint operator in the Hilbert space H. Resolvent set, spectrum, essential spectrum and discrete spectrum of the operator A are denoted by p, a, aess and a disc, respectively (see [21]).

We define the numbers

Emin(A) = inf (A : A e aess (A)},

Emax(A) = sup (A : A e aess(A)}.

We call the number Emin(A) (Emax(A)) the lower (the higher) boundary of the essential spectrum of A.

Let tt i = [a, b]Vl, ^2 = [c, d]V2 and ko, ki, are continuous functions on tti x tt2, tt2 x tt2, tt1 x tt2 respectively, ko is real function, k1 (x, s,y) = k!(s,x,y),k2(x,t,y) = k2(x,y,t). We define the linear self-adjoint bounded operator H in the Hilbert space L2(tt1 x tt2) by rule

H = To - (Ti + T2). (2)

We set

T = Ti + T2.

For the essential spectrum of the operators H and T the equalities

aess(T) = a(Ti) U a(T2),

aess(H) = a (To - Ti) U a(To - T2) (3)

are held (see [22],[10]).

Let ki(x, s,y) = a = const > 0,k2(x,t,y) = ft = const > 0 in the model (2). Then at the Emin(H) = 0 the Efimov's effect (the existence infinite numbers of eigenvalues below the lower boundary Emin(H) of the essential spectrum) in the model (2) was demonstrated by S. Albeverio, S.N.Lakaev, Z.I. Muminov [11] and Rasulov T.Kh. [12].

We study the existence Efimov's effect in the model (2) in the case Emin(H) = 0. Consider this problem for the function k0(x,y) of the form k0(x,y) = u(x)u(y) and ko(x,y) = u(x) + u(y).

Let u(x) and v(y) be a continuous nonnegative function on tti and tt2, respectively and suppose ki(x,s,y) = ki(x, s), k2(x,t,y) = k2(y,t). We define the self-adjoint compact integral operators Qi : L2(tti) ^ L2(tti) and Q2 : L2(tt2) ^ L2(tt2) by the following equalities

Qi<p(x) = ki(x,s)<p(s)ds, Q^(y) = \ k2(y,t)^(t)dt

Jqi jQ2

and suppose that Q1 > 9, Q2 > 9.

We define by V1 and V2 multiplication on the space L2(Q1) and L2(Q2) are acting by the following formulas

V p(x) = u(x)p(x), V2^(y) = v(y)^(y). We consider the operators Hk ,k = 1,2 in the Friedrichs model:

H = V! - Q1, (4)

H2 = V2 - Q2. (5)

3. Efimov's effect for PIO

Let u(x) > 0, x e Qi,v(y) > 0, y e Q2 and u-1({0}) = 0, v-1({0}) = 0. Theorem 3.1. Let k0(x, y) = u(x) + v(y), u(x) > 0, v(y) > 0 and Q1 > 9, Q2 > 9.

(a) the aess(H) = a(Ho) iff the adlsc(H) = 0;

(b) if <Jdisc(H) = 0, then adisc(Hi) = 0 and v^scH) = 0;

(c) if V disc (H) = 0, then Emin(H) = inf {A : A e Vdisc(Hl) U VdiscH)};

(d) Efimov's effect exists in the model (2) iff Efimov's effect exists in Friedrich's model (4) and vdisc(H2) = 0 or Efimov's effect exists in Friedrich's model (5) and vdisc(H1) = 0.

Proof. We define the operator W = H1 ® E + E ® H2 on the space L2(Q1) ® L2(Q2). For the spectrum of the operator W we have [18]

v(W ) = v(H1) + v(H2).

But, the operators W and H is unitary equivalent (see [10]), i.e. W = H. Consequently, that

v(H ) = v(H1) + v (H2). (6)

Also, if we define the operators A1 and A2 by

A1 = H1 0 E + E 0 V2, A2 = V1 0 E + E 0 H2 we see A1 = To - Tu A2 = To - T2. Thus we have

v(To - T1) = v(H1) + v(V2), v(To - T2) = v(V1) + v(H2). Then, by the equality (6) for the essential spectrum of the operator H we obtain

Vess(H) = (v(H1) + v(V2)) U (v(V1) + v(H2)). (7)

On the other hand

Vess(Hk) = v(Vk), k =1, 2. (8)

Now, from the equalities (6), (7) and (8) it follows proof of theorem 1. □

Corollary 3.1. Let be k0(x,y) = u(x) + v(y) and k1(x, s,y) = k1(x, s), k2(x,t,y) = k2(y,t).

a) for the existence of Efimov's effect in model (2) it is necessary, that dim(Ran(Qi)) = to or dim(Ran(Q2)) = to;

b) if dim(Ran(Qi)) < to and dim(Ran(Q2)) < to, then Efimov's effect is absent in model (2).

Suppose, that ko(x,y) > 0,0 e Ran(ko) and Tfc > 0,k = 1, 2. Let No(H) be the number of all eigenvalues (with account multiplicity) below the lower boundary of the essential spectrum in model (2), i.e.

No(H) = £ nn (A),

Aeadisc(H), \<£min(H)

where nH (A) - the multiplicity of the eigenvalue A of the operator H and N(T) is the number of all eigenvalues(with account multiplicity) of the discrete spectrum of operator T, i.e.

N (T) = £ nT (A).

A£&disc(T )

Theorem 3.2. Let the relation

To > (Emin(H) + Emax(T)) ' E

is hold, where E - identical operator. Then

No(H) < N(T).

Proof. We have

aess(Emax(T) ■ E - T) = {£ : £ = Emax(T) - A, A e aess(T)}.

Then

aess (Emax(T) ■ E - T) C [0, TO)

and

0 e aess(Emax(T) - T) , N(Emax(T) ■ E - T) = N(T).

Hence it follows

Emin(T) Emin(Emax(T) ' E T) 0.

Let the inequality

To > (Emin(H) + Emax(T)) ■ E

hold. Consequently, we obtain

Emax(T) ■ E - T < H - Emin(H) ■ E

Then by lemma 2.1 [23] we have

Vk(Emax(T) ■ E - T) < Vk(H -Emin(H) ■ E), k e(1,2, ..., N(T) + 1}. (10)

where (vk(A)} - the set of all eigenvalues of operator A below the lower boundary of the essential spectrum, which was constructed by the mini-max principle. By theorem 2.1 [23] we have

VN(T)+i (Emax (T) ' E - T) = 0.

Therefore, from inequality (10)

Vn(T)+k(H -Emin(H) ■ E) = 0, k e N U(0},

i.e. the number of negative eigenvalues of operator H - Emin(H) ■ E taking into account multiplicity, is no more than N(T). Consequently, the number of eigenvalues of the operator H below the lower boundary Emin(H) of the essential spectrum, also will be no more than N(T), i.e.

No(H) < N(T). □

Corollary 3.2. Let be N(T) < to. If

Emin(H) + Emax(T) < 0,

then Efimov's effect is absent in model (2).

Let u(x) and v(y) be nonnegative continuous functions on tti and tt2, respectively. Suppose, that u-i({0}) = (xmin}, v-i((0}) = (ymin}.

Theorem 3.3. Let be ko(x,y) = u(x)v(y). Then

(a) the equality Emin(H) = - max(||Qi||, ||Q211} holds;

(b) if dim(Ran(Qi)) < to and dim(Ran(Q2)) < to, then Efimov's effect is absent in model (2).

Proof. (a) We define the family (Wi(a)}aen2 of self-adjoint operators in Fredrich's model on the space L2(tti) :

Wi(a)^(x) = u(x)v(x)^(x) - Qi^(x).

We set

ai = (A e (-to, 0) : for some ao e tt2 the number A is eigenvalue of the operator Wi(ao)}. Then by the theorem 6 from [24] we have

a (To - Ti) = ao U ai,

where ao = a(To). However,

((To - Ti)/,/) > -(Ti/,/) > - ||Qi||, / e L2(tti x tt2),

because ||Ti| = ||Qi|. Conversely, Wi(ymin) = -Qi, i.e. the number -|Qi| is eigenvalue of the operator Wi(ymin). Consequently, -|Qi| e ai, i.e. -|Qi| e a(To - Ti). Then we have Emin(To - Ti) = ||Qi|. Analogously, for the operator To - T2 we obtain, that Emin(To - T2) = - ||Q2||. Finally, from (3) follows, that Emin(H) = -max{||Qi||, ||Q2|}.

(b) By statement (a) of theorem 3.3 we have Emin(H) = - max(|Qi|, ||Q2|}. However, Emax(T) = max(|Qi|, ||Q2|} (see [25]) Consequently, the condition of theorem 2 is satisfied. Still, by theorem 3.1 from [25] (also see [18]) we have

N(T) = nQi (p) • nQ2 (q)>

ess (t ),

P €&disc(Ql), q €&disc(Q 2)

where aess(T) = a(Qi) U a(Q2). Then, from the inequality dim(Ran(Qi)) < ro and dim(Ran(Q2)) < ro, we obtain N(T) < ro. Consequently, by theorem 3.2 Efimov's effect is absint in model (2). □

Remark 3.1. The author's previous work [14] showed the existence of Efimov's effect in the case dim(Ran(Qi)) = 1 and dim(Ran(Q2)) = ro in the model (2) for the

Emin(H) = 0 .

4. The examples

Example 4.1. On Q = [0,1]v, we consider the functions

(2).

u(x) = xiai • ... • xvav, v(y) = ylix • ... • y

where ak > > 0, k,j e {1, ..,v}. In the space L2(Q2), we define the operators

Tof (x,y) = (u(x)+ v(y))f (x,y), Tif (x,y)= exp(\x - s\)f (s,y)ds,

T2f (x,y)= exp(\y - t\)f (x,t)dt, Jn

where

\x\ = xi2 + ... + xu 2. If ai + ... + av > 2v and + ... + > 2v, then Efimov's effect exist for operator

We define subsets An(n e N) and Bn(n e N) by the following way

An = {t e Q: 0 < ti < 1, i = 1, ..., v}, n e N,

n

Bn = {t e Q: 0 < ti < , i = 1, ..., v}, n e N. n - i n+ V

We have Bn c An, n e N. We set

Gn = An\Bn, n e N.

Then Gn n Gm = 0 at n = m and Gn c Oi (9) n Q, where Or(9) - the open ball with radius

n

r in center 9 e Rv(9 - the zero element of the space Rv). For the Lebesque measure ^(Gn) of the set Gn we obtain

V(Gn) = V(An) - V(Bn) = -1 - , \ > -1 = „vfi}v , -, v > 1

nv (n + 1)v nv nv + 1 nv (nv +1) 2n2v

Efimov's effect for partial integral operators of Fredholm type for all n e N, n > 2. On the other hand

' 1

sup u(t) tec- \ n

ai + ...

ne N.

Consequently, if ai + ... + av > 2v, then the following inequality

sup u(t) < v(Gn) inf ki(t,u), n e N\{1}.

tecn t,ueGn

holds, i.e. the condition of theorem 4.1 from [26] is satisfied. So, by theorem 4.1 operator Hi = Vi - Qi has an infinite number of negative eigenvalues.

Analogously, we show that, at the + ... + > 2v the operator H2 = V2 - Q2 has infinite number of negative eigenvalues.

Therefore, by the theorem 3.1 Emin(H) = 0 and Efimov's effect exists for the PIO H.

Remark 4.1. The statement of theorem 4.1 from [26] holds for the set Q = [0,1]v for arbitrary v e N. In work [26] proof was given for the simple case v =1. The proof of theorem 4.1 [26] for the case v > 2 is analogous to the case v =1.

Example 4.2. We consider the sequence po = 0, pi = 1/2, pn = pn-i + 1/2n, n e N. We set

Pn Pn-1 TNT

qn = -~-, n G N.

On [0,1], we define the function

u(x) =

0, if x G [0,1/2], uo(x), if x G [o, 1/2],

where uo(x) = ^ ^n(x),

neN

rK(x)

Pk - x

if x G [Pk, ?re+l],

if x G [qK+i,PK+i], Pk+i - qK+i

0, if x G [Pk,Pk+i],

Pk - qK+l Pk+i - x

¿1 = 1, ¿n < ( ^

n > 2.

n

In the space L2[0,1], we consider the sequence of orthonormal functions

^n(y) = 2(n+1)/2 sin£n(y), n G N,

where

n

&(y)

We define the kernel

Pk - Pk-1

-(y - Pk-i^ if y G [Pk-i,Pk],

0,

if y G [Pk-i,Pk].

k2(y,t) = ^ i 33 ) ^(y)^n(t).

raGN ^

(11)

Series (11) converges uniformly in the square [0,1]2. Hence, the integral operator Q2, defined by its kernel k2(y,t), is self-adjoint and positive in L2[0,1]. It is clear, that dim(Ran(Qi)) = 1 and dim(Ran(Q2)) = to.

In the space L2([0,1]2), we consider the model

where

2

H = Ho - (7Ti + T2), y > 2,

(12)

/ (s,y)dS

Then

T2f (x,y)^ k2(y,t)f (x,t)dt.

J0

Emin(H)= Emin(Hc - (7T1 + T2))

and there exists Efimov's effect for operator (12) below the lower boundary Emin(H) of the essential spectrum [14].

Y

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