Analysis of the spectrum of a 2×2 operator matrix. Discrete spectrum asymptotics Текст научной статьи по специальности «Математика»

Analysis of the spectrum of a 2 x 2 operator matrix. Discrete spectrum asymptotics

We consider a 2 x 2 operator matrix , p > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of We describe the new branches of the essential spectrum of via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of consists the union of at most three bounded closed intervals and their location is studied. For the critical value p0 of the coupling constant p we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of In this case, an asymptotic formula for the discrete spectrum of is found.

Keywords: operator matrix, bosonic Fock space, coupling constant, dispersion function, essential and discrete spectrum, Birman-Schwinger principle, spectral subspace, Weyl creterion. Received: 30 December 2019 Revised: 17 February 2020

1. Introduction and statement of the problem

It is well-known that [1], if H is a bounded linear operator in a Hilbert space H and a decomposition H = Hi ®H2 into two Hilbert spaces Hi, H2 is given, then H always admits a block operator matrix representation

with bounded linear operators Hij : Hj ^ Hi, i,j = 1,2. In addition, H = H* if and only if Hii = Hi*i, i =1, 2 and H2i = H*2. Such operator matrices often arise in mathematical physics, e.g., in quantum field theory, condensed matter physics, fluid mechanics, magnetohydrodynamics and quantum mechanics. One of the special class of 2 x 2 block operator matrices is the Hamiltonians acting in the one- and two-particle subspaces of a Fock space. It is related with a system describing three-particles in interaction without conservation of the number of particles in Fock space. Here, off-diagonal entries of such block operator matrices are annihilation and creation operators.

Operator matrices of this form play a key role for the study of the energy operator of the spin-boson Hamiltonian with two bosons on the torus. In fact, the latter is a 6 x 6 operator matrix which is unitarily equivalent to a 2 x 2 block diagonal operator with two copies of a particular case of H on the diagonal, see e.g. [2]. Consequently, the location of the essential spectrum and finiteness of discrete eigenvalues of the spin-boson Hamiltonian are determined by the corresponding spectral information on the operator matrix H. We recall that the spin-boson model is a well-known quantum-mechanical model which describes the interaction between a two-level atom and a photon field. We refer to [3] and [4] for excellent reviews from physical and mathematical perspectives, respectively. Independently of whether the underlying domain is a torus Td or the whole space Rd, the full spin-boson Hamiltonian is an infinite operator matrix in Fock space for which rigorous results are very hard to obtain. One line of attack is to consider the compression to the truncated Fock space with a finite number N of bosons, and in fact most of the existing literature concentrates on the case N < 2. For the case of Rd there are some exceptions, e.g. [5,6] for arbitrary finite N and [7] for N = 3, where a rigorous scattering theory was developed for small coupling constants.

For the case when the underlying domain is a torus, the spectral properties of some versions of H were investigated in [8-11]. An important problem of the spectral theory of such matrix operators is the infiniteness of the number of eigenvalues located outside the essential spectrum. We mention that, the infiniteness of the discrete eigenvalues below the bottom of the essential spectrum of the Hamiltonian in Fock space, which has a block operator matrix representation, and corresponding eigenvalue asymptotics were discussed in [8]. These results were obtained using the machinery developed in [12] by Sobolev.

In the present paper we consider a 2 x 2 operator matrix (p > 0 is a coupling constant) related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the

T. H. Rasulov, E. B. Dilmurodov

Faculty of Physics and Mathematics, Bukhara State University M. Ikbol str. 11, 200100 Bukhara, Uzbekistan rth@mail.ru, elyor.dilmurodov@mail.ru

DOI 10.17586/2220-8054-2020-11-2-138-144

H

Hii Hi2

H21 H22

number of particles. This operator acts in the direct sum of one- and two-particle subspaces of the bosonic Fock space and it is related with the lattice spin-boson Hamiltonian [2,13]. We find the critical value j0 of the coupling constant j, to establish the existence of infinitely many eigenvalues lying in both sides of essential spectrum of AM0 and to obtain an asymptotics for the number of these eigenvalues.

We point out that the latter assertion seems to be quite new for the discrete models and similar result have not been obtained yet for the three-particle discrete Schrodinger operators and operator matrices in Fock space. In all papers devoted to the infiniteness of the number of eigenvalues (Efimov's effects), the situation on the neighborhood of the left edge of essential spectrum are discussed, see for example [8-10,14-16]. Since the essential spectrum of the three-particle continuous Schrodinger operators [12,17,18] and standard spin-boson model with at most two photons [19,20] coincides with half-axis [k; the main results of the present paper are typical only for lattice case, and they do not have analogs in the continues case.

Now, we formulate the problem. Let T3 be the three-dimensional torus, the cube (-n,n}3 with appropriately identified sides equipped with its Haar measure. Let L2(T3) be the Hilbert space of square integrable (complex) functions defined on T3 and L2((T3)2) be the Hilbert space of square integrable (complex) symmetric functions defined on (T3)2. Denote by H the direct sum of spaces Hi := L2 (T3) and H2 := L2((T3)2), that is, H := Hi ©H2. The spaces H1 and H2 are called one- and two-particle subspaces of a bosonic Fock space Fs(L2(T3)) over L2(T3), respectively.

Let us consider a 2 x 2 operator matrix acting in the Hilbert space H as:

A11 fj*Ai2 ^A*i2 A22

with the entries

(Anf1)(k) = w1(k)f1(k), (Ai2/2)(fc) = J f2(k, s)ds,

t3

(A22f2)(k,p) = w2(k,p)f2(k,p), fi e Hi, i = 1,2. Here, ^ > 0 is a coupling constant, the functions w1( ) and w2(:, •) have the form

wi(k) := e(k) + y, W2(k,p) := e(k) + ^(k + p)) + e(p)

with y e R and the dispersion function e( ) is defined by:

3

e(k) := ^(1 - cos ki), k = (k1, k2,k3) e T3, (1.1)

i= 1

A12 denotes the adjoint operator to A12 and

(Al2fi)(k,p) = 1(fi(k) + fi(p)), fi e Hi.

Under these assumptions, the operator is bounded and self-adjoint.

We remark that the operators Ai2 and Ai2 are called annihilation and creation operators [21], respectively. In physics, an annihilation operator is an operator that lowers the number of particles in a given state by one, a creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator.

2. Faddeev's equation and essential spectrum of A^

In this section, we obtain an analog of the Faddeev type integral equation for eigenvectors of and investigate the location and structure of the essential spectrum of AM.

Throughout the present paper we adopt the following conventions: Denote by a( ), aess( ) and adisc( ), respectively, the spectrum, the essential spectrum, and the discrete spectrum of a bounded self-adjoint operator.

Let H0 := C. To study the spectral properties of the operator we introduce a family of bounded self-adjoint operators (generalized Friedrichs models) A^(k),k e T3 which acts in H0 © Hi as 2 x 2 operator matrices:

< Aoo(k) -^Aci

-J=ASi Aii(k)

with matrix elements:

Aoo(k)/o = wi(k)/o, (Aoi/i) = y /(t)dt,

T3

(Aii(k)/2)(p) = W2(k,p)/i(p), /i G Hi, i =1,2.

From the simple discussions it follows that CTess(AM(k)) = [m(k), M(k)], where the numbers m(k) and M(k) are defined by:

m(k) := min w2(k,p), M(k) := max w2(k,p). (2.1)

peT3 peT3

For any k G T3 we define an analytic function: I(k; •) in C \ aess(AM(k)) by

I(k ; z) := f ■ J

dt

W2 (k, t) — z

T3

Then the Fredholm determinant associated to the operator AM(k) is defined by:

p2

AM(k ; z) := wi(k) — z — I(k ; z), z G C \ aeSS(AM(k)).

A simple consequence of the Birman-Schwinger principle and the Fredholm theorem implies that for the discrete spectrum of AM(k), the equality:

^disc(AM(k)) = {z G C \ [m(k); M(k)] : AM(k; z) = 0}

holds.

We set:

m := min w2(k,p), M := max w2(k,p),

k,pe t3 k,pet3

AM := U CTdisc(AM(k)), := [m; M] U AM.

keT3

For each p > 0 and z G C \ we define the integral operator TM(z) acting in the Hilbert spaces L2(T3) by

(TM(z)g)(p) =

g(t)dt

2AM(p; z) J W2(p,t) — z'

T3

2

p

The following theorem is an analog of the well-known Faddeev's result for the operator and establishes a connection between eigenvalues of and TM(z).

Theorem 2.1. The number z e C \ is an eigenvalue of the operator AM if and only if the number X =1 is an eigenvalue of the operator TM(z). Moreover, the eigenvalues z and 1 have the same multiplicities.

We point out that the integral equation g = TM(z)g is an analog of the Faddeev type system of integral equations for eigenfunctions of the operator AM and it is played crucial role in the analysis of the spectrum of AM. For the proof of Theorem 2.1 we show the equivalence of the eigenvalue problem Af = zf to the equation g = TM(z)g.

The following theorem describes the location of the essential spectrum of the operator AM by the spectrum of the family of generalized Friedrichs models AM(k).

Theorem 2.2. For the essential spectrum of AM, the equality o-ess(AM) = holds. Moreover, the set consists of no more than three bounded closed intervals.

The inclusion c aess(AM) in the proof of Theorem 2.2 is established with the use of a well-known Weyl creterion, see for example [11]. An application of Theorem 2.1 and analytic Fredholm theorem (see, e.g., Theorem VI.14 in [18]) proves inclusion aess(AM) C

In the following we introduce the new subsets of the essential spectrum of AM.

Definition 2.3. The sets AM and [m; M] are called two- and three-particle branches of the essential spectrum of AM, respectively.

The definition of the set AM and the equality

y [m(k); M(k)] = [m; M]

fceT3

together with Theorem 2.2 give the equality

^SS(AM) = y a(A^(k)). (2.2)

fceT3

Here the family of operators AM(k) have a simpler structure than the operator AM. Hence, in many instances, (2.2) provides an effective tool for the description of the essential spectrum.

Using the extremal properties of the function w2( , ), and the Lebesgue dominated convergence theorem one can show that the integral I(0; 0) is finite, where 0 := (0,0,0) e T3, see [22,23]. For the next investigations we introduce the following quantities

m0(y) := fY (I(0,0))-i/2 for y > 0; m0(y) := f 24 - 2y (I(0,0))-i/2 for y < 12.

Since T3 is compact, and the functions AM(-; 0) and AM(-; 18) are continuous on T3, there exist points k0,ki e T3 such that the equalities

max AM(k; 0) = AM(kc; 0), min AM(k; 18) = AM(ki; 18)

fee t3 fee t3

hold.

Let us define the following notations:

Y0 := (12^ - £(kc)) (W ^

Y0 V I(0;0) ( 0V V I(0;0)

Yi := (18 - £(ki)) (1 - I(ki;18)

We denote:

I (0;0)

Ed := min|AM n (-rc;0]} ; £(2) := max |AM n (-rc;0]} ; E(3) := min |AM n [18; rc)} ; E(4) := max |AM n [18; rc)} .

We formulate the results, which precisely describe the structure of the essential spectrum of AM. The structure of the essential spectrum depends on the location of the parameters ^ > 0 and y e R.

Theorem 2.4. Let p = p°(y), with y < 12. The following equality holds

|>i; E2]U[0;18], f Y < Y°; ^eSS(AM) = < [Ei; 18], if yo < Y < 6;

[[0; 18], if 6 < y < 12.

Theorem 2.5. Let p = p°(y), with y > 0. The following equality holds:

( [0; 18], if 0 < y < 6;

^eSS(AM)= H0; £W], if 6 < y < Yi;

[ [0; 18] UE3); Ef], if Y>Yi.

The proof of these two theorems are based on the existence conditions of the eigenvalue zM(k) of the operator AM( ) and the continuity of zM( ) on its domain.

3. Birman-Schwinger principle and discrete spectrum asymptotics of the operator AM

Let us denote by rmin(AM) and rmax(AM) the lower and upper bounds of the essential spectrum aess(AM) of the operator AM, respectively, that is,

Tmin(AM) := minaeSS(AM), Tmax(AM) := maxaess(AM).

For an interval A c R, Ea(Am) stands for the spectral subspace of corresponding to A. Let us denote by } the cardinality of a set and by N(a,b) (AM) the number of eigenvalues of the operator AM, including multiplicities, lying in (a, b) c R \ CTess(AM), that is,

N(a,b)(AM) := dimE(0j6)(AM). For a A G R, we define the number n(A, ) as follows

n(A, AM) := sup{dimF : (AMu, u) > A, u G F cH, ||u|| = 1}.

The number n(A, AM) is equal to the infinity if A < maxaess(AM); if n(A, AM) is finite, then it is equal to the number of the eigenvalues of bigger than A. By the definition of N(a;b) (AM), we have

N(-TO;z)(AM) = n(-z, -AM), -z > -Tmin(AM),

N(z;+TO)(AM) = n(z, AM), z >

Tmax

In our analysis of the discrete spectrum of AM, the crucial role is played by the compact operator TM(z), z G R \ [Tmin(AM); Tmax(AM)] in the space L2(T3) as integral operator

(T„(z)g)(p) = . p2 i . -, for z < Tmin(AA

( M )g)(P) 2^AM(p; z) J TA^)(w2(p,t) - z), min(

T3

(T„(z)g)(p) =--, ^ i , g(t)dt-, for z > Tmax(AA

( M )g)(P) 2V-A>; z)J V-AM(i; z)(w2(p,t) - z), max( m)

T3

The following lemma is a realization of the well-known Birman-Schwinger principle for the operator (see [8]).

Lemma 3.1. For z e R \ [rmin(AM); rmax(AM)] the operator TM(z) is compact and continuous in z and

(AM) = n(1,TM(z)) for z < Tmin(AM),

(AM) = n(1,TM(z)) for z>

This lemma can be proven quite similarly to the corresponding result of [8]. Let S2 being the unit sphere in R3 and

Sr : L2((0,r),ao) ^ ¿2«0, r), a°), r> 0, a° = ¿2(S2)

be the integral operator with the kernel

S(t; y) = 8n2V6 5cos(hy) + t, y = x - x', x,x' e (0, r), t = (£, n), £,,n e S2.

For X > 0, define

U(X) = 1 lim r-in(X,Sr).

2 r^w

The existence of the latter limit and the fact U (1) > 0 shown in [12].

From the definitions of the quantities m°(y) and m°(y), it is easy to see that ^°(6) = ^°(6). We set := ^°(6). We can now formulate our last main result.

Theorem 3.2. The following relations hold:

tt(adisc(AM0) n (-rc, 0)) = tt(adiSc(AM0) n (18, rc)) = rc;

lim n(-^,.)(AMo) = lim N(z,w)(Ao) = u(1). (3.1)

z^o | log |z|| z\is | log |z - 181| w v y

Clearly, by equality (3.1), the infinite cardinality of the parts of discrete spectrum of AM0 in (-rc;0) and (18; +rc) follows automatically from the positivity of U(1).

Acknowledgements

The authors thank the anonymous referee for reading the manuscript carefully and for making valuable suggestions.

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