On the number of eigenvalues of the family of operator matrices Текст научной статьи по специальности «Математика»

ON THE NUMBER OF EIGENVALUES OF THE FAMILY OF OPERATOR MATRICES

M.I. Muminov1, T.H. Rasulov2

1Universiti Teknologi Malaysia, Faculty of Science, Department of Mathematical Sciences, 81310 UTM Johor Bahru, Malaysia 2Bukhara State University, Faculty of Physics and Mathematics, 11 M. Ikbol str., Bukhara, 200100, Uzbekistan 1 [email protected], [email protected]

PACS 02.30.Tb

We consider the family of operator matrices H(K), K e T3 := ( — n; n]3 acting in the direct sum of zero-, one-and two-particle subspaces of the bosonic Fock space. We find a finite set A c T3 to establish the existence of infinitely many eigenvalues of H(K) for all K e A when the associated Friedrichs model has a zero energy resonance. It is found that for every K e A, the number N(K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim N(K, z)| log |z|| 1 = U0 with 0 < U0 < <x, independently on the cardinality of A. Moreover, we show that for any K e A the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.

Keywords: operator matrix, bosonic Fock space, annihilation and creation operators, Friedrichs model, essential spectrum, asymptotics. Received: 3 June 2014

1. Introduction

One of the important problems in the spectral theory of Schrodinger operators and operator matrices in Fock space is to study the finiteness or infiniteness (Efimov's effect) of the number of eigenvalues located outside the essential spectrum. The Efimov effect for the three-particle continuous Schrodinger operator has been discussed in [5]. A rigorous mathematical proof of the existence of this effect was originally carried out by Yafaev [14] and then many works devoted to this subject, see for example [12,13].

It was shown in [1,2,7] that for the three-particle discrete Schrodinger operator HM(K), the Efimov effect exists only for the zero value of the three-particle quasi-momentum (K = 0) and for some value ^ = > 0 of the interaction energy of two particles. Moreover, the operator H^(K) has only a finite number of eigenvalues for all sufficiently small nonzero values of K and ^ > 0. An asymptote analogous to [12,13] was obtained in [1,2] for the number of eigenvalues of HM(K).

In all above mentioned papers devoted to the Efimov effect, the systems where the number of quasi-particles is fixed have been considered. In solid-state physics theory [10], quantum field theory [6] and statistical physics [9] some important problems arise where the number of quasi-particles is finite, but not fixed.

In the present note, we consider the family of 3 x 3 operator matrices H(K), K e T3, associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set

A c T3 and under some smoothness assumptions on the parameters of a family of Friedrichs models h(k), k G T3, we obtain the following results: if h(0) has a zero energy resonance, then for the number N(K, z) of eigenvalues of H(K) lying on the left of z, z < 0, we establish the asymptotics N(K, z) ~ U0| log |z|| with 0 < U0 < ro for all K G A.

We show the finiteness of negative eigenvalues of H(K) for K G A, if the operator h(0) has a zero eigenvalue or a zero is the regular type point for h(0) with h(0) > 0.

We point out that the operator H(K) has been considered before in [3,4,8] for K = 0 and n =1, where the existence of Efimov's effect has been proven. Moreover, similar asymptotics for the number of eigenvalues was obtained in [3].

2. Family of 3 x 3 operator matrices and main results

We denote by T3 the three-dimensional torus, the cube (—n,n]3 with appropriately identified sides equipped with its Haar measure. Let Ho := C be the field of complex numbers, Hi := L2(T3) be the Hilbert space of square integrable (complex) functions defined on T3 and H2 := Ls2((T3)2) be the Hilbert space of square integrable (complex) symmetric functions defined on (T3)2. The spaces H0, H1 and H2 are called zero-, one- and two-particle subspaces of a bosonic Fock space Fs(L2(T3)) over L2(T3), respectively.

Let us consider the following family of 3 x 3 operator matrices H(K), K G T3 acting in the Hilbert space H := Ho © Hi © Hs as :

( Hoo (K)

H(K) :=

H

oi

Hoi Hii(K)

H

12

0 \

Hi2 Hss(K) y

with the entries:

Hoo(K )fo = Wo(K )fo, Hoifi

Vo(s)/i(s)ds, (Hii(K )/i)(p) = wi(K; p)/i(p),

T3

(Hi2 /2)(P) = ^S^s^S^ (H22(K)/2)(P,q) = w2(K; P^^SCP,?),

T3

where f G Hi, i = 0,1, 2; w0(-) and Vj(-), i = 0,1 are real-valued bounded functions on T3, the functions wi(-; ■) and w2(-; ■, ■) are defined by the equalities:

Wi(K;p) := lie(p) + /2^(K - p) + 1, W2(K;p, q) := /ie(p) + lie(q) + /2^(K - p - q),

respectively, with 1i, 12 > 0 and

e(q) := ^(1 - cos(nq(i))), q = (q(i), q(2), q(3)) G T3, n G N.

i=i

Here, Hj (i < j) denotes the adjoint operator to Hij and

(H0ifo)(p) = Vo(p)fo, (H*2/i)(p,q) = 1(vi(p)fi(q) + vi(q)fi(p)), fi G Hi, i = 0,1.

Under these assumptions, the operator H(K) is bounded and self-adjoint.

We remark that the operators H0i and Hi2 resp. H0i and Hj*2 are called annihilation resp. creation operators [6], respectively. In this note, we consider the case where the number of annihilations and creations of the particles of the considering system is equal to 1. It means that Hij = 0 for all |i - j| > 1.

0

where

We denote by aess(-) and adisc(-), respectively, the essential spectrum, and the discrete spectrum of a bounded self-adjoint operator.

To study the spectral properties of the operator H(K), we introduce a family of bounded self-adjoint operators (Friedrichs models) h(k), k e T3, which acts in Ho © Hi as follows:

h(b\ ■— I hoo(k) h01 h(k):=l h0i hii (k)

hoo(k)fo = (¿2^(k) + 1)fo, hoi fi = J vi(s)fi(s)ds,

T3

(hii(k)fi)(q) = Ek(q)fi(q), Ek(q) := he(q) + ^(k - q). It is easily to seen that aess(h(0)) = [0; 6(1i + 12)].

The following theorem describes the location of the essential spectrum of operator H(K) by the spectrum of the family h(k) of Friedrichs models.

Theorem 2.1. For the essential spectrum of H(K), the equality

aess(H(K)) = U |adisc(h(K - p)) + he(p)} U [mK; Mk]

peT3

holds, where the numbers mK and MK are defined by:

mK := min w2(K;p,q) and MK := max w2(K;p,q).

p,qeT3 p,qeT3

Let us consider the following subset of T3 :

A := |(p(i),p(2),p(3)) : p(i) e (0, ±-n; ±4n;...; ±-4 U n„, i = 1, 2, 3

n n n

where

, _ . n — 2, if n is even d n — J {^1, if n is even n — 1, if n is odd n ' | 0, if n is odd

Direct calculation shows that the cardinality of A is equal to n3. It is easy to check that for any K e A, the function w2(K; ■, ■) has non-degenerate zero minimum at the points of A x A, that is, mK — 0 for K e A.

The following assumption we needed throughout the note: the function vi(-) is either even or odd function on each variable and there exists all second order continuous partial derivatives of v1(-) on T3.

Let us denote by C(T3) and L1(T3) the Banach spaces of continuous and integrable functions on T3, respectively.

Definition 2.2. The operator h(0) is said to have a zero energy resonance if the number 1 is an eigenvalue of the integral operator given by:

(G*)(q) — í ds, * e C(T3)

2(11 + ¿2)7 e(s)

T3

and at least one (up to a normalization constant) of the associated eigenfunction * satisfies the condition *(p') — 0 for some p' e A. If the number 1 is not an eigenvalue of the operator G, then we say that z — 0 is a regular type point for the operator h(0).

We note that in Definition 2.2, the requirement for the existence of an eigenvalue 1 of G corresponds to the existence of a solution of h(0)/ = 0 and the condition -0(p') = 0 for some p' G A implies that the solution / of this equation does not belong to H0 ® Hi. More precisely, if the operator h(0) has a zero energy resonance, then the solution of G^ = ^ is equal to vi(-) (up to constant factor) and the vector / = (/0, /i), where

/0 = const = 0, /i(q) =--^ Vi(q)/o-, (2.1)

/0 = , /i(q) V2(h + feMq),

obeys the equation h(0)/ = 0 with /i G Li(T3) \ L2(T3). If the operator h(0) has a zero eigenvalue, then the vector / = (/0, /i), where /0 and /i are defined by (2.1), again obeys the equation h(0)/ = 0 and /i G ¿2(T3).

Denote by Tess(K) the bottom of the essential spectrum of H(K) and by N(K, z) the number of eigenvalues of H(K) on the left of z, z < Tess(K).

Note that if the operator h(0) has either a zero energy resonance or a zero eigenvalue, then for any K G A and p G T3 the operator h(K - p) + /ie(p)/ is non-negative, where I is the identity operator in H0 ® Hi. Hence Theorem 2.1 and equality mK = 0, K G A imply that Tess(K) = 0 for all K G A.

The main results of the present note are as follows.

Theorem 2.3. Let K G A and one of the following assumptions hold:

(i) the operator h(0) has a zero eigenvalue;

(ii) h(0) > 0 and a zero is the regular type point for h(0).

Then the operator H(K) has a finite number of negative eigenvalues.

Theorem 2.4. Let K G A. If the operator h(0) has a zero energy resonance, then the operator H(K) has infinitely many negative eigenvalues accumulating at zero and the function N (K, ■) obeys the relation:

lim N(K,z) = Uo, 0 < Uo < ro. (2.2)

| log |z||

Remark 2.5. The constant U0 does not depend on the functions v0(-), vi(-) and the cardinality of the set A. It is positive and depends only on the ratio /2//i.

Remark 2.6. Clearly, by equality (2.2), the infinite cardinality of the negative discrete spectrum of H (K) follows automatically from the positivity of U0.

3. Sketch of proof of the main results

For any k G T3, we define an analytic function A(k; ■) (the Fredholm determinant associated with the operator h(k)) in C \ [Emin(k); Emax(k)] by:

A(k ; z) := 12 e(k) + 1 - z --' iv ;

2 J Ek(s) - z'

T3

where the numbers Emin(k) and Emax(k) are defined by

Emin(k) := min Ek(q) and Emax(k) := max Ek(q).

q€T3 q€T3

Set

S(K) := U l^disc(h(K - p)) + /ie(p)} U [mK; Mk].

peT3

Let us consider 2 x 2 block operator matrix T(K, z), z e C \ E(K) acting on Ho © Hi

as

T(Kz)-=( TToo(K,z) Toi(K,z) ( , ):V -Tio(K, z) Tii(K, z)

with the entries Tj (K, z) : Hj ^ Hj, i, j = 0,1 :

Too(K, z)go = (1 + z - Wo(K))go, ?oi(K, z) = -Hoi;

vo(p)go

(TTio(K,z)go)(p) = -(Tii(K,z )gi )(p)

A(K - p; z- Zie(p))' vi(p) f vi(s)gi(s)ds

2A(K - p; z - /^(p)) J w2(K; p, s) - z'

T3

gi e Hj, i = 0,1.

The following lemma is an analog of the well-known Faddeev's result for the operator H(K) and establishes a connection between eigenvalues of H(K) and T(K, z).

Lemma 3.1. The number z e C \ E(K) is an eigenvalue of the operator H(K) if and only if the number A =1 is an eigenvalue of the operator T(K, z). Moreover, the eigenvalues z and 1 have the same multiplicities.

The inclusion E(K) C aess(H(K)) in the proof of Theorem 2.1 is established with the use of a well-known Weyl criterion. An application of Lemma 3.1 and analytic Fredholm theorem (see, e.g., Theorem VI.14 in [11]) proves inclusion aess(H(K)) C E(K).

To find conditions which guarantee for the finiteness or infiniteness of the number of eigenvalues of H(K), K e A, we establish in which cases the bottom of the essential spectrum of h(0) is a threshold energy resonance or eigenvalue.

Lemma 3.2. (i) The operator h(0) has a zero eigenvalue if and only if A(0;0) = 0 and vi(q/) = 0 for all q' e A;

(ii) The operator h(0) has a zero energy resonance if and only if A(0;0) = 0 and vi(q/) = 0 for some q' e A.

Since A(K - p; z - /ie(p)) > 0 for any K,p e T3 and z < Tess(K), one can define a symmetric version of the operator T(K, z) for z < Tess(K), which is important in our analysis of the discrete spectrum of H(K), K e T3. So, we consider the self-adjoint compact 2 x 2 block operator matrix T(K, z), z < Tess(K) acting on Ho © Hi as follows:

T(K z) = / Too(K,z) Toi(K,z)

( ,z):^ T?i(K,z) Tii(K,z) with the entries Tj(K, z) : Hj ^ Hj, i, j = 0,1 :

vo(s)gi(s)ds

Too(K, z)go = (1 + z - Wo(K))go, Toi(K, z)gi = -

J A(K - s ; z - /ie(s))'

T3

frp fUr , W X Vi(p) f vi(s)gi(s)ds

(Tii(K,z)gi)(p) '

VA(K - p; z - /ie(p)^ ^/A(K - s ; z - he(s))(w2(K;p, s) - z)'

T3

gj e Hj, i = 0,1.

To prove Theorem 2.3, first we show N(K, z) = n(1,T(K, z)) (so-called BirmanSchwinger principle for the operator H(K)), where n(1,A) is the number of the eigenvalues

(counted multiplicities) of the compact operator A bigger than 1. Then, under the conditions of Theorem 2.3, we prove that the operator T(K, z) is continuous from the left up to z = 0 and T(K, 0) is a compact operator. Using the Weyl inequality,

n(Ai + A2, Ai + A2) < n(Ai, Ai) + n(A2, A2)

for the sum of compact operators Ai and A2, and for any positive numbers Ai and A2, we have

n(1,T(K, z)) < n(1/2,T(K, 0)) + n(1/2,T(K, z) - T(K, 0))

for all z < 0. Hence, lim N(K, z) = N(K, 0) < n(1/2,T(K, 0)) < ro.

The study of the behavior of T(K, z), K G A, that is, proof of Theorem 2.4, is based on the analysis of the behavior of A(K - p; z - /ie(p)) as z ^ -0 and |p - p'| ^ 0 for K,p' G A. Set

Ao := {q' G A : vi(q') = 0}.

Lemma 3.3. Let the operator h(0) have a zero energy resonance and K,p' G A. Then, the following decomposition:

A(K - p; z - w)) = (E v^i w'p - p^2 - §

q'eA 0 '

+ O(|p - p'|2) + O(|z|),

holds for |p - p'| ^ 0 and z ^ -0.

By applying Lemma 3.3, we single out the principal part of the operator T(K, z) K G A as z ^ -0, which is unitarily equivalent to the compact integral operator SR, R = 1/21 log |z|| in L2((0, R), L2(S2)) with the kernel

S(y,t):=^ ('i + '2)2 1 ,

4n2 + 21i 12 ('i + '2) cosh y + ¿21

where S2 be the unit sphere in R3; y = x - x', x, x' G (0, R) and t = (£, n) is the inner product of the arguments £, n G S2.

The eigenvalue asymptotics for the operator SR have been studied in detail by Sobolev [12], by employing an argument used in the calculation of the canonical distribution of Toeplitz operators.

Acknowledgments

This work was supported by the IMU Einstein Foundation Program. T. H. Rasulov wishes to thank the Berlin Mathematical School and Weierstrass Institute for Applied Analysis and Stochastics for the invitation and hospitality.

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