Threshold analysis for a family of 2 × 2 operator matrices Текст научной статьи по специальности «Математика»

Threshold analysis for a family of 2 x 2 operator matrices

T. H. Rasulov, E. B. Dilmurodov

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

DOI 10.17586/2220-8054-2019-10-6-616-622

We consider a family of 2 x 2 operator matrices A»(k), k € T3 := (—n, n]3, p > 0, acting in the direct sum of zero- and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z3, interacting via annihilation and creation operators. We find a set A := {k(1),..., k(8)} C T3 and a critical value of the coupling constant p to establish

necessary and sufficient conditions for either z = 0 = min ae33 (A„(k)) (or z = 27/2 = max ae33 (A„(k)) is a threshold eigenvalue or a

fceT3 feel3

virtual level of A» (k(i)) for some k(i) € A.

Keywords: operator matrices, Hamiltonian, generalized Friedrichs model, zero- and one-particle subspaces of a Fock space, threshold eigenvalues, virtual levels, annihilation and creation operators. Received: 19 October 2019 Revised: 13 November 2019

1. Introduction

Operator matrices are matrices where the entries are linear operators between Banach or Hilbert spaces, see [1]. One special class of operator matrices are Hamiltonians associated with the systems of non-conserved number of quasi-particles on a lattice. In such systems the number of particles can be unbounded as in the case of spin-boson models [2,3] or bounded as in the case of "truncated" spin-boson models [4-7]. They arise, for example, in the theory of solid-state physics [8], quantum field theory [9] and statistical physics [4,10].

The study of systems describing n particles in interaction, without conservation of the number of particles is reduced to the investigation of the spectral properties of self-adjoint operators acting in the cut subspace H(n) of the Fock space, consisting of r < n particles [4,9,10]. The perturbation of an operator (the generalized Friedrichs model which has a 2 x 2 operator matrix form acting in H(2)), with discrete and essential spectrum has played a considerable role in the study of spectral problems connected with the quantum theory of fields [9].

One of the most actively studied objects in operator theory, in many problems of mathematical physics and other related fields is the investigation of the threshold eigenvalues and virtual levels of block operator matrices, in particular, Hamiltonians on a Fock space associated with systems of non-conserved number of quasi-particles on a lattice. In the present paper, we consider a family of 2 x 2 operator matrices AM(k), k G T3 := (-n, n]3, ^ > 0 (so - called generalized Friedrichs models) associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z3, interacting via creation and annihilation operators. They are acting in the direct sum of zero-particle and one-particle subspaces of a Fock space. The main goal of the paper is to give a thorough mathematical treatment of the spectral properties of this family in three dimensions. More exactly, we find a set A := {k(1),..., k(8)} c T3 and prove that for a i G {1,2,..., 8} there is a value ^ of the parameter p such that only for p = fa the operator AM(0) has a zero-energy resonance, here 0 = min CTess(AM(0)) and the operator AM(k(i)) has a virtual level at the point z = 27/2 = max aeSS(AM(k(i))), where 0 := (0,0,0) G T3 and k(i) G A. We point out that a part of the results is typical for lattice models; in fact, they do not have analogs in the continuous case (because its essential spectrum is half-line [E; see for example [4]).

We notice that threshold eigenvalue and virtual level (threshold energy resonance) of a generalized Friedrichs model have been studied in [11-14]. The paper [15] is devoted to the threshold analysis for a family of Friedrichs models under rank one perturbations. In [16] a wide class of two-body energy operators h(k) on the d-dimensional lattice Zd, d > 3, is considered, where k is the two-particle quasi-momentum. If the two-particle Hamiltonian h(0) has either an eigenvalue or a virtual level at the bottom of its essential spectrum and the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups, then it is shown that for all nontrivial values k, k = 0, the discrete spectrum of h(k) below its threshold is non-empty. These results have been applied to the proof of the existence of Efimov's effect and to obtain discrete spectrum asymptotics of the corresponding Hamiltonians. We note that above mentioned results are discussed only for the bottom of the essential spectrum. The threshold

eigenvalues and virtual levels of a slightly simpler version of AM(k) were investigated in [17], and the structure of the numerical range are studied using similar results. In [18], the essential spectrum of the family of 3 x 3 operator matrices H(K) is described by the spectrum of the family of 2 x 2 operator matrices. The results of the present paper are play important role in the investigations of the operator H(K), see [12].

The plan of this paper is as follows: Section 1 is an introduction to the whole work. In Section 2, a family of 2 x 2 operator matrices are described as bounded self-adjoint operators in the direct sum of two Hilbert spaces and its spectrum is described. In Section 3, we discuss some results concerning threshold analysis of a family of 2 x 2 operator matrices.

We adopt the following conventions throughout the present paper. Let N, Z, R and C be the set of all positive integers, integers, real and complex numbers, respectively. We denote by T3 the three-dimensional torus (the first Brillouin zone, i.e., dual group of Z3), the cube (—n, n]3 with appropriately identified sides equipped with its Haar measure. The torus T3 will always be considered as an abelian group with respect to the addition and multiplication by real numbers regarded as operations on the three-dimensional space R3 modulo (2nZ)3.

Denote by a( ), aess( ) and <rdisc(0, respectively, the spectrum, the essential spectrum, and the discrete spectrum of a bounded self-adjoint operator.

2. Family of 2 x 2 operator matrices and its spectrum

Let L2 (T3) be the Hilbert space of square-integrable (complex-valued) functions defined on the three-dimensional torus T3. Denote H by the direct sum of spaces Ho := C and Hi := L2(T3), that is, H := Ho © H1. We write the elements f of the space H in the form f = (f0, f1) with f0 g H0 and f1 G H1. Then for any two elements f = (f0, f1) and g = (g0, g1), their scalar product is defined by

(f,g) := f0g0 + J f1(t)gK)d*.

T3

The Hilbert spaces H0 and H1 are zero- and one-particle subspaces of aFock space F(L2(T3)) over L2(T3), respectively, where

F(L2(T3)) := C © L2(T3) © L2((T3)2) © • • • © ¿2((T3)") © • • • . In the Hilbert space H we consider the following family of 2 x 2 operator matrices

A00(k)

MQ1 An(k)

AM(k) :=

where A^(k) : Hi ^ Hi, i = 0,1, k G T3 and A01 : H1 ^ H0 are defined by the rules

A00(k)f0 = W0(k)f0, A01f1 = J v(t)f1(t)dt, (An(k)f1)(p) = W1(k,p)f1(p).

T3

Here fi G Hi, i = 0,1; ^ > 0 is a coupling constant, the function v() is a real-valued analytic function on T3, the functions w0(-) and •) have the form

W0(k) := e(k) + 7, w1(k,p) := e(k) + e(k + p) + e(p)

with 7 g R and the dispersion function e( ) is defined by

3

e(k) := ^(1 - cos ki), k = (k1, k2, k3) G T3. (2.1)

i=1

Under these assumptions the operator matrix AM(k) is a bounded and self-adjoint in H.

We remark that the operators A01 and aq1 are called annihilation and creation operators [9], 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.

Let A0(k) := AM(k)|M=0. The perturbation AM(k) - A0(k) of the operator A0(k) is a self-adjoint operator of rank 2. Therefore, in accordance with the invariance of the essential spectrum under the finite rank perturbations [19], the essential spectrum aess(AM(k)) of AM(k) fills the following interval on the real axis

^eSS(AM(k)) = [m(k),M (k)],

where the numbers m(k) and M(k) are defined by

m(k) := min w1(k,p), M(k) := max w1(k,p). (2.2)

peT3 peT3

Remark 2.1. We remark that the essential spectrum of AM(n), n := (n, n, n) G T3 is degenerate to the set consisting of the unique point {12} and hence we can not state that the essential spectrum of AM(k) is absolutely continuous for any k G T3.

For any fa > 0 and k G T3 we define an analytic function AM(k; •) in C \ aess(AM(k)) by

AM(k ; z) := wo(k) - z - fa2 / v?(t)dt , z G C \ aeSS(AM(k)).

J Wl(k, t) — z

T3

Usually the function AM(k; •) is called the Fredholm determinant associated to the operator matrix AM(k). The following statement establishes connection between the eigenvalues of the operator AM(k) and zeros of the function AM(k; •), see [11,14].

Lemma 2.2. For any fa > 0 and k G T3 the operator AM(k) has an eigenvalue zM(k) G C \ aess(AM (k)) if and only if AM(k; zM(k)) =0.

From Lemma 2.2 it follows that

^disc(AM(k)) = {z G C \ aeSS(AM(k)) : AM(k ; z) = 0}.

Since the function AM(k; •) is a monotonically decreasing function on (—to; m(k)) and (M(k); +to), for fa > 0 and k G T3 the operator AM(k) has no more than 1 simple eigenvalue in (—to; m(k)) and (M(k); +to).

Let A := {k = (ki, k?, k3) : k G {— 2n/3,2n/3}, i = 1,2,3}. Since the set A c T3 consists 8 points for a convenience we rewrite the set A as A = {k(1),k(2),...,k(8)}.

It is easy to verify that the function w1( , •) has a non-degenerate minimum at the point (°, °) G (T3)2, 0 := (0,0,0) and has non-degenerate maximum at the points of the form (k(i), k(i)) g (T3)2, i = 1,..., 8, such that

min W1(k,p)= w1(0, 0)=0, max w1(k,p) = w2(k(i), k(i)) = 27/2, i = 1,..., 8.

fc,peT3 fc,peT3

Simple calculations show that

^eSS(AM(0)) = [0; 12];

15 27

aeSS(AM(k(i))) = [Y;y], i = 1,..., 8.

Therefore,

27

min CTess(AM(k)) = 0, max ffess(AM(k)) = —

kE T3 kE T3 2

3. Threshold eigenvalues and virtual levels

In this Section, we prove that for any i e {1,..., 8} there is a value fa¿ of the parameter (coupling constant) fa such that only for fa = fa¿ the operator AM(0) has a virtual level at the point z = 0 (zero-energy resonance) and the operator (k(i)) has a virtual level at the point z = 27/2 under the assumption that v(°) = 0 and v (k(i)) = 0. For the case v(0) = 0 and v(k(i)) = 0 we show that the number z = 0 (z = 27/2) is a threshold eigenvalue of (0)

(AM(k(i))).

Denote by C(T3) and Li(T3) the Banach spaces of continuous and integrable functions on T3, respectively.

Definition 3.1. Let y = 0. The operator AM(°) is said to have a virtual level at z = 0 (or zero-energy resonance), if the number 1 is an eigenvalue of the integral operator

(G^)(p) = fa2VYP) / * e C(T3)

T3

and the associated eigenfunction *(■) (up to constant factor) satisfies the condition *(0) = 0.

Definition 3.2. Let y = 9 and i G {1,..., 8}. The operator AM(fc(i)) is said to have a virtual level at z = 27/2, if the number 1 is an eigenvalue of the integral operator

(G-V)(p) = W-9J £(k(i) +t) + £(t) - 9, * G C(T)

T3

and the associated eigenfunction y(-) (up to constant factor) satisfies the condition ^>(fc(i)) = 0.

Using the extremal properties of the function e( ), and the Lebesgue dominated convergence theorem, we obtain that there exist the positive finite limits

lim f ^ = r fm.;

z-m-oj e(t) - z J e(t) '

T3 T3

f v2(t)dt f v2(t)dt

lim ' — '

-1/2

for 7 < 9, i = 1,..., 8.

z-9+oJ z - e(k(i) + t) - e(t) J 9 - e(k(i) +t) - e(t)'

T3 T3

For the next investigations, we define the following quantities

( \ -1/2 Mi(y):= ^ U ^ j for y > 0;

,(i)(7) := (i__

M( (Y):V9 Y 1/ 9 - £(k(i) + t) - e(t)

T3

Let y. G (0; 9) be an unique solution of mi (y) = m^ (y) . It follows immediately that

:= 9 (2 i v2(t)dt + [ v2(t)dt j 1 f v2(t)dt

Yi : I J 9 - e(k(i) +1) - e(t)+y J e(t) .

T3 T3 T3

In the following, we compare the values of mKy) and m( (y) depending on y g (0; 9).

Remark 3.3. Let i G {1,..., 8}. By the definition of the quantities mi (y) and Mr^Y) one can conclude that if Y G (0; Yi), then mi(y) < M^y); if Y = Yi, then ^(y) = M^y); if Y G (Yi;9), then mi(y) > M^y).

From the Definition 3.1 (resp. 3.2) we obtain that the number 1 is an eigenvalue of GM (resp. G^) if and only if m = My) (resp. m = M(i)(Y)).

(i)

We notice that in the Definition 3.2, the requirement of the presence of an eigenvalue 1 of G^) corresponds to the existence of a solution of the equation AM(fc(i))f = (27/2)f and the condition ^(k(i)) = 0 implies that the solution f = (/0, /1) of this equation does not belong to H. More exactly, if the operator (k(i)) has a virtual level at z = 27/2, then the vector-function f = (f0, f1), where

fof'<">=-,(«.) rf -9• (3-i)

satisfies the equation AM(k(i))f = (27/2)f and f G MT3) \ L2 (T3) (see assertion (i) of Theorem 3.4).

If the number z = 27/2 is an eigenvalue of the operator AM(k(i)) then the vector-function f = (f0, f1), where f0 and f1 are defined in (3.1), satisfies the equation AM(k(i))f = (27/2)f and f1 G L2(T3) (see assertion (ii) of Theorem 3.4).

The same assertions are true for the operator AM(0) at the point z = 0.

Henceforth, we shall denote by C1, C2, C3 different positive numbers and for each S > 0, the notation Us (p0) is used for the S-neighborhood of the point p0 G T3 :

Us(po) := {p G T3 : |p - po| < S}.

Now we formulate the first main result of the paper.

Theorem 3.4. Let y < 9 and i G {1,..., 8}.

(i) The number z = 27/2 is an eigenvalue of the operator AM(k(i)) ifandonly if fa = fa^Y) and v(k(i)) = 0;

(ii) The operator AM(k(i)) has a virtual level at the point z = 27/2 ifandonly if fa = fa^Y) and v(k(i)) = 0.

Proof. Suppose y < 9 and i G {1,..., 8}.

(i) "Only If Part". Let the number z = 27/2 be an eigenvalue of AM(k(i)) and f = (/0,/1) G H be an associated eigenvector. Then f0 and f 1 are satisfy the system of equations

(Y — 9)fo + fa J v(t)/1 (t)dt = 0;

T3

fav(p)fo + (e(k(i) + p) + £(p) — 9)f1(p) = 0. (3.2)

This implies that f0 and f1 are of the form (3.1) and the first equation of system (3.2) yields AM(k(i); 27/2) = 0, therefore, fa = fa^Y).

Now let us show that f1 G L2(T3) if and only if v(k(i)) = 0. Indeed, if v(k(i)) = 0 (resp. v(k(i)) = 0), from analyticity of the function v( ) it follows that there exist C1, C2, C3 > 0, 6i g N and S > 0 such that

C1|p — k(i)|ei < |v(p)| < C2|p — k(i)|e*, p G Us(k(i)), (3.3)

respectively

|v(p)| > C3, p G T3 \ Us(k(i)). (3.4)

Since the function e(k(i) + p) + e(p) has an unique non-degenerate maximum at the point k(i) G T3 there exist C1, C2, C3 > 0 and S > 0 such that

C1|p — k(i)|2 < |e(k(i) + p)+ e(p) — 9| < C2|p — k(i)|2, p G Us(k(i)), (3.5)

|e(k(i) + p) + e(p) — 9| > C3, p G T3 \ Us(k(i)). (3.6)

We have

'|f1(t)|2dt = fa2|fo|2 ^ V2(t)dt

(e(k(i) +t) + e(t) — 9)2

T3 Us (k(i))

+ ,,2| f |2 I v2(t)dt (37)

+falf0| J (£(k(i) +t) + e(t) — 9)2. (37)

T3\U (fcM)

Let v(k(i)) = 0. Then by (3.3) and (3.5) for the first summand on the right-hand side of (3.7) we have

f v2(t)dt ^ f |t — k(i)|2^ dt

' < C I ^-¡-p^— < +TO.

J (e(k(i) + t) + e(t) — 9)2 " 1 7 |t — k(i)|4

U (k(i)) U (fc(i))

It follows from the continuity of v( ) on a compact set T3 and (3.6) that

f v2(t)dt ^ f

J (g(k(i) + t)+g(t) — 9)2 < C1 J dt< +TO.

T3\Us(k(i)) T3\US (fcW)

So, in this case f1 G ¿2(T3).

For the case v(k(i)) =0 there exsist the numbers S > 0 and C1 > 0 such that |v(p)| > C1 for any p g Us (k(i)). Then from (3.5) we obtain

/|f1(t)|2dt > C1 J ^^ = +to.

T3 Us(k(i))

Therefore, f1 G L2(T3) ifandonly if v(k(i)) = 0.

"If Part". Suppose that fa = fa^Y) and v(k(i)) = 0. It is easy to verify that the vector-function f = (f0, f1) with f0 and f1 defined in (3.1) satisfies the equation AM(k(i))f = (27/2)f. We proved above that if v(k(i)) = 0, then f1 G L2(T3).

(ii) "Only If Part". Suppose that the operator AM(k(i)) has a virtual level at z = 27/2. Then by Definition 3.2 the equation

*(p) = -Y—9J £(k(i) + t) + e(t) — 9, * G C(T) (3.8)

T3

has a nontrivial solution ^ g C(T3), which satisfies the condition = 0.

This solution is equal to the function v(p) (up to a constant factor) and hence

2 f v2(t)dt

AM(k(i), 27/2)= y - 9 -

e(k(i) + t) + e(t) - 9

T3

that is, m = m^(y).

"If Part". Let now m = Mr (y) and v(k(i)) = 0. Then the function v G C(T3) is a solution of (3.8), and consequently, by Definition 3.2 the operator AM(k(i)) has a virtual level at z = 27/2. □

The following result may be proved in much the same way as Theorem 3.4.

Theorem 3.5. Let y > 0.

(i) The operator AM(0) has an zero eigenvalue if and only if m = mí(y) and v(0) = 0;

(ii) The operator AM(0) has a zero energy resonance if and only if m = mí(y) and v(0) = 0.

Since ^í(yí) = M^ (yí), setting := mí (y¿), from Theorems 3.4 and 3.5 we obtain the following

Corollary 3.6. Let y G (0; 9) and i G {1,..., 8}.

(i) The operator AM(0) has a zero eigenvalue and the number z = 27/2 is an eigenvalue of AM(k(i)) iff m = Mi and v(0) = v(k(i)) = 0;

(ii) The operator AM(0) has zero-energy resonance and the operator AM(k(i)) has a virtual level at the point z = 27/2

iff m = Mi, v(0) = 0 and v(k(i)) = 0;

(iii) The operator AM(0) has a zero eigenvalue and the operator AM(k(i)) has a virtual level at the point z = 27/2 iff M = Mi, v(0) = 0 and v(k(i)) = 0;

(iv) The operator AM(°) has a zero-energy resonance and the number z = 27/2 is an eigenvalue of AM(k(i)) iff M = Mi, v(0) = 0 and v(k(i)) = 0.

Next we will consider some applications of the results. Denote by H2 := L|((T3)2) the Hilbert space of square integrable (complex) symmetric functions defined on (T3)2. In the Hilbert space Hi © H2 we consider a 2 x 2 operator matrix

. .= f An V2mai2 ^ where Aij : Hj ^ Hi, i = 1,2 are defined by the rules

l v^ma^ A22

(An/i)(k) = wi(k)fi(k), (Ai2f2)(k) = J v(t)/2(k,t)dt,

T3

(A22/2)(k,p) = wi(k,p)/2(k,p) /i G Hj, i = 1, 2. Here : Hi ^ H2 denotes the adjoint operator to Ai2 and

(Ai2/i)(k,p) = 2(v(k)/i(p) + v(p)/i(k)), /i G Hi. Under these assumptions the operator is bounded and self-adjoint.

The main results of the present paper plays crucial role in the study of the spectral properties of the operator matrix AM. In particular, the essential spectrum of can be described via the spectrum of AM(fc) the following equality holds

^eSS(AM) = [0; 27/2] U J adiSc(AM(k)).

fceT3

Since the operator AM(fc) has at most 2 simple eigenvalues, the set aess(AM) consists at least one and at most three bounded closed intervals, for similar results see [7].

Using Theorems 3.4 and 3.5 one can investigate [14] the number of eigenvalues of and find its discrete spectrum asymptotics. We note that the case

v(p) = VM = const, wi(k,p) = e(fc) + ^(^(k + p)) + e(p) is studied in [13], and it is shown that the bounds min o-ess(AM(0)) and max CTess(AM(n)) are only virtual levels. This

kET3 kET3

paper generalizes the results of the paper [13] and it is proved that these bounds are threshold eigenvalues or virtual levels depending on the values of the function v( ).

Acknowledgements

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

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