UNIVERSALITY OF THE DISCRETE SPECTRUM ASYMPTOTICS OF THE THREE-PARTICLE SCHRöDINGER OPERATOR ON A LATTICE Текст научной статьи по специальности «Математика»

Universality of the discrete spectrum asymptotics of the three-particle Schrodinger operator on a lattice

Mukhiddin I. Muminov1, Tulkin H. Rasulov2

1Faculty of Scince, Universiti Teknologi Malaysia (UTM) 81310 Skudai, Johor Bahru, Malaysia 2Faculty of Physics and Mathematics, Bukhara State University M. Ikbol str. 11, 200100 Bukhara, Uzbekistan mmuminov@mail.ru, rth@mail.ru

PACS 02.30.Tb DOI 10.17586/2220-8054-2015-6-2-280-293

In the present paper, we consider the Hamiltonian H(K), K e T3 := (—n; n]3 of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set A c T3 such that for all values of the total quasi-momentum K e A the operator H(K) has infinitely many negative eigenvalues accumulating at zero. 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.

Keywords: Three-particle Schrodinger operator, zero-range pair attractive potentials, Birman-Schwinger principle, the Efimov effect, discrete spectrum asymptotics. Received: 18 January 2015

1. Introduction

We are going to discuss the following remarkable phenomenon of the spectral theory of the three-particle Schrodinger operators, known as the Efimov effect: if a system of three particles interacting through pair short-range potentials is such that none of the three two-particle subsystems has bound states with negative energy, but at least two of them have a zero energy resonance, then this three-particle system has an infinite number of three-particle bound states with negative energy accumulating at zero.

The Efimov effect was discussed in [1] for the first time. Since then, this problem has been studied on a physical level of rigor in [2,3]. A rigorous mathematical proof for the existence of Efimov's effect was originally carried out in [4], and subsequently, many works have been devoted to this subject, see for example [5-9]. The main result obtained by Sobolev [7] (see also [9]) is an asymptotics of the form U0|log|z|| for the number N(z) of eigenvalues on the left of z, z < 0, where the coefficient U0 does not depend on the two-particle potentials va and is a positive function of the ratios m1/m2 and m2/m3 of the masses of the three particles.

In models of solid state physics [10-12] and also in lattice quantum field theory [13], one considers discrete Schroodinger operators, which are lattice analogs of the three-particle Schrodinger operator in continuous space. The presence of Efimov's effect for these operators was demonstrated at the physical level of rigor without a mathematical proof for a system of three identical quantum particles in [10,11].

In the continuous case [14] (see also [12,15]), the energy of the center-of-mass motion can be separated out from the total Hamiltonian, that is, the energy operator can be split into the sum of the center-of-mass motion and the relative kinetic energy so that the

three-particle bound states are eigenvectors of the relative kinetic energy operator. Therefore, Efimov's effect either exists or does not exist for all values of the total momentum simultaneously.

In lattice terms, the center-of-mass corresponds to a realization of the Hamiltonians as fibered operators, that is, as the direct integral of a family of operators H(K) depending on the values of the total quasi-momentum K e T3 := (—n; n]3 (see [12]). In this case, a bound state is an eigenvector of the operator H(K) for some K e T3. Typically, this eigenvector depends continuously on K. Therefore, Efimov's effect may exist only for some values of K e T3.

The presence of the Efimov effect for three-particle discrete Schrodinger operators was proved in [16-18] and asymptotic formulas for the number of eigenvalues were obtained in [16,17], which are analogous to the results of [7,9].

In the present paper, we consider a system of three arbitrary quantum particles on the three-dimensional lattice interacting via zero-range potentials with the dispersion

3

function of the form e(p) = ^(1 — cos(np(i))) with n > 1. We denote by A the set of points

i=1

of T3 where the function e(-) takes its (global) minimum. If at least two of the two-particle operators have a zero energy resonance and third one is non-negative, then we prove that for all K e A, the three-particle discrete Schrodinger operator H(K) has infinitely many negative eigenvalues accumulating at zero. Moreover, for any K e A, we establish the asymptotic formula

lim N(K; z)| log |z||-1 = Uo (0 < Uo < ro),

where N(K; z) is the number of eigenvalues of H(K) lying on the left of z, z < 0.

It is surprising that the asymptotics for N(K; z) is the same for all K e A and is stable with respect to the number n. Recall that in all papers devoted to Efimov's effect for lattice systems, the existence of this effect has been proved only for the zero value of the quasi-momentum (K = 0) and for the case n =1. In [19], for all non-trivial values of total quasi-momentum (K = 0), the finiteness of the discrete spectrum of a system of three bosons on a lattice was proven when the corresponding two-particle operator has a zero energy resonance.

The plan of this paper is as follows: Section 1 is an introduction to the whole work. In Section 2, the Hamiltonians of two- and three-particle systems are described as bounded self-adjoint operators in the corresponding Hilbert spaces and the main result of the paper is formulated. In Section 3, we discuss some results concerning threshold analysis of the two-particle operator ha(k). In Section 4, we give a modification of the Birman-Schwinger principle for H(K), K e T3. In Section 5, we obtain an asymptotic formula for the number of negative eigenvalues of H(K), K e A.

Throughout the present paper, we adopt the following conventions: For each 5 > 0, the notation Us(p0) := {p e T3 : |p — p0| < 5} stands for a 5-neighborhood of the point p0 e T3. The subscripts a, y are pair-wisely different and takes values from {1,2,3}.

2. Description of the three-particle operator

Let Z3 be the three-dimensional lattice and /2((Z3)m) be the Hilbert space of square-summable functions on (Z3)m, m = 2,3. The free Hamiltonian H0 of a system of three arbitrary quantum mechanical particles on Z3 in the coordinate representation is usually

associated with the following bounded self-adjoint operator on the Hilbert space /2((Z3)3) : (HoV0(Xl,X2,£3) = [H (s)V>(£i + S,X2,X3)+ £2 (s)^/)(Xi,X2 + «,£3)+ £3 (s)i(£1, £2, X3 + s)],

seZ3

where Ha(-), a = 1,2,3 are dispersion functions describing the particle transition from a site to a neighboring site defined by:

Va(s) :=

3

— as s = 0;

ma 1

as s = ±ne,, i = 1, 2, 3;

2ma 0 otherwise.

Here, ma > 0 are different numbers, having the meaning of a mass of the particle a, a = 1, 2, 3, the elements e», i = 1,2, 3 are unit orts on Z3 and n is a fixed positive integer with n > 1.

It is easily seen that the function Ha(-) is even on Z3.

The three-particle Hamiltonian H of the quantum-mechanical three particle systems with two-particle interactions Hg7, ^,7 = 1,2,3 in the coordinate representation is a bounded perturbation of the free Hamiltonian HH0:

H = Ho — V1 — V2 — V3, where Va, a = 1,2,3 are multiplication operators on the Hilbert space /2((Z3)3)

(Ha^H)(x1,X2,X3) = H37(X^ — X7)i(£1, £2, £3) = ^a^x7i(»1, »2, £3) , i e /2 ((Z3)3) . Here, > 0 is the interaction energy of the particles ft and 7, 5X/3Xy is the Kronecker delta.

It is clear that the three-particle Hamiltonian H is a bounded self-adjoint operator on the Hilbert space /2((Z3)3).

Similarly, as we introduced H, we introduce the corresponding two-particle Hamil-tonians ha, a = 1,2,3 as bounded self-adjoint operators on the Hilbert space /2 ((Z3)2):

ha = ha - V

where

vaV (xe, £7) = ^ [v(s)^v(xe + s, £7) + v(s)^v(xe, £7 + s)]

(VaV) (£в,£7) = x7V(Xe,£7), V € l2((Z3)2).

Let us rewrite our operators in the momentum representation. We denote by T3 the three-dimensional torus, the cube (—n,n]3 with appropriately identified sides and L2 ((T3)m) be the Hilbert space of square integrable (complex) functions defined on (T3)m, m = 1,2,3.

Let Fm : L2 ((T3)m) ^ /2((Z3)m), m = 2, 3 be the discrete Fourier transform. The three-particle Hamiltonian in the momentum representation is given by the bounded self-adjoint operator on the Hilbert space L2 ((T3)3) as follows H = F-1ííF3. Introducing the total quasi-momentum K € T3 the operator H can be decomposed into von Neumann direct integrals of the family of bounded self-adjoint operators H(K), K € T3. The operator H(K), K € T3 is called the three-particle discrete Schrodinger operator, which is unitarily

equivalent (see [16-18]) to the family of bounded self-adjoint operators H(K), K e T3, acting on the Hilbert space L2 ((T3)2) according to the formula:

H(K) = Ho(K) - Vi - V2 - V3,

where H0(K) is the multiplication operator by the function:

Ek(p,q) := £i(p) + £2(q) + £3(K - p - q),

where

1 3 £a(p) :=-e(p), e(p) := (1 - cos(np(i))) ,

i=i

and

(V/)(p,q) = ^/f(p,s)ds (V2f)(p,q) = ^/f(s,q)ds

T3 T3

^3 r

(V3f)(p,q) = ^y f(s,p + q-

T3

Similarly, the study of the spectral properties of the ha = F2_1haF2 can be reduced to the study of the spectral properties of the family of bounded self-adjoint operators ha(k), k e T3, corresponding to the two-particle lattice Hamiltonians on the Hilbert space L2(T3):

h«(k) = h^(k) - .

The non-perturbed operator hO,(k) is the multiplication operator on L2(T3) by the function:

E(C)(p) := ^(p)+ (k - p), £< Y, a, £,Y = 1, 2, 3.

The perturbation is an integral operator of rank one on L2(T3):

(v«/)(p) = /f (s)ds

T3

Therefore, by the Weyl theorem, the continuous spectrum acont(h«(k)) of the operator ha(k) coincides with the spectrum of a(hO,(k)) of hO,(k). More specifically:

acont(h«(k)) = [^¡^(k); EiOX(k)'

where

ESin(k) := minEka)(p) and £¡1^) := max4a)(p).

pO3 pO3

3. Formulation of the main results

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

The following theorem, [17,18], describes the location of the essential spectrum of the operator H(K).

Theorem 3.1. For the essential spectrum of H(K) the following equality holds:

3

aess(H(K))= U U i^disc(h«(K - p))+ £«(p)}u [Emin(K); Emax(K)], (3.1)

a=1p€T3

where

Emin(K) := min Ek(p,q) and Emax(K) := max Ek(p,q).

p,q€T3 p,q€T3

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

where

Let us consider the following subset of T3

2 4 -

:—n; ±—n;...; ±— - - -

-__i - — 2, if - is even; ^^ n__[ {n}, if - is even;

:= - — 1, if - is odd, n = } 0, if - is odd.

Direct calculation shows that the cardinality of A is equal to -3. It is easy to verify that for any K e A, the function EK(■, ■) has non-degenerate zero minima at the points of A x A, that is, Emin(K) = 0 for K e A.

Since 0 = (0,0,0) e A, the definition of the functions Ek«(■) and EK(■, ■) imply the identities h«(0) = h«(k) and H(0) = H(K) for all k,K e A.

Let C(T3) and L1(T3) be the Banach spaces of continuous and integrable functions on T3, respectively. Let Ga be the integral operator on C(T3) with the kernel:

mg m7 1

Ga(p, s)

(2n)3 mg + mY e(s)

Definition 3.2. The operator ha(0) is said to have a zero energy resonance if the number 1 is an eigenvalue of the operator Ga. If the number 1 isn't an eigenvalue of the operator Ga, then we say that z = 0 is a regular-type point for the operator ha(0).

We note that in Definition 3.2 the requirement of the existence of the eigenfunction <Pa e C(T3) corresponding to the eigenvalue 1 of Ga corresponds to the existence of a solution of ha(0)/a = 0, and this solution does not belong to L2(T3). More precisely, if the operator ha(0) has a zero energy resonance, then the function:

/«(p) = ^a(p)(e(p))

1

is a solution (up to a constant factor) of the Schrodinger equation ha(0)/a = 0 and /a e L1(T3) \ L2(T3) (see Lemma 4.4).

We set:

^ :=8n3m + m ( i , a = 1, 2, 3.

mgm7 \ J e(s) /

T3

Simple calculation shows that the operator ha(0) has a zero energy resonance if and only if ^a = ^ (see Lemma 4.2).

For K e T3, let us 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) lying on the left of z, z < Tess(K). It is clear that N(0; z) = N(K; z) for any K e A.

Since the operator ha(0) has no negative eigenvalues for all < ^ (see Lemma 4.3), the operator ha(0) is non-negative for all < ^ Then, by Theorem 1 of [21], the operator ha(k) is non-negative for all < ^a and k e T3. Hence, the assertion Emin(K) = 0, K e A implies Tess(K) = 0 for K e A and < ^

The main result of the present paper is given in the following theorem.

Theorem 3.3. Assume = ^a, ^ = and < ^Y. Then the operator H(0) has infinitely many negative eigenvalues accumulating at zero and the function N(0; ■) obeys the relation:

lim ^^ = Uo, 0 < Uo < rc. (3.2)

| log |z||

Remark 3.4. The constant U0 does not depend on the interaction energies a = 1,2,3; it is positive and depends only on the ratios m^/ma, a = a,^ = 1, 2, 3 between the masses.

Remark 3.5. Clearly, by equality (3.2), the infinite cardinality of the negative discrete spectrum of H(0) follows automatically from the positivity of U0.

Remark 3.6. It is surprising that the asymptotics (3.2) don't depend on the cardinality of A, that is, these asymptotics are the same for all n e N. Since A|n=1 = {0} in fact, Theorem 3.3 was proved in [17] for n =1.

4. Threshold analysis of the two-particle operator ha(k)

In this section, we study the spectral properties of the two-particle discrete Schrodinger operator ha(k).

For any > 0, k e T3 and z e C\acont(ha(k)) we define the function (the Fredholm determinant associated with the operator ha(k)):

^a f ds

A«(fc ; z) := 1 -

(2n)3i (s) - z'

T3 k x '

Note that the function Aa(-; ■) is analytic in T3 x (C \ acont(ha(k))). The following lemma is a simple consequence of the Birman-Schwinger principle and the Fredholm theorem.

Lemma 4.1. The number z e C \ acont(ha(k)) is an eigenvalue of the operator ha(k), k e T3 if and only if Aa(k ; z) = 0.

We remark that from the definition of E(a)(-), it follows that Aa(0 ;0) = Aa(k; 0) for k e A. k

Lemma 4.2. The following statements are equivalent:

(i) the operator ha(0) has a zero energy resonance;

(ii) Aa(0 ; 0) = 0;

(iii) ^a = <

For the proof of Lemma 4.2, see Lemma 5.3 of [17].

Lemma 4.3. The operator ha(0) has no negative eigenvalues for all < ^

Proof. Since the function Aa(0; ■) is decreasing on (-ro;0), we have

Aa(0 ; z) > Aa(0 ; 0) (4.1)

for all z < 0. Definition of ^ implies Aa(0; 0) > 0 for all < ^ Hence by inequality (4.1) we have Aa(0; z) > 0 for any < ^ and z < 0. By Lemma 4.1, it means that the operator ha(0) has no negative eigenvalues for all < ^ ^

In the sequel, we denote by C1,C2, C3 different positive numbers and for 5 > 0 we

set:

Ts := T3 \ U Us(pi).

p'eA

Lemma 4.4. If ha(0) has a zero energy resonance, then the function /a(p) = (p)(e(p))-1 obeys the equation ha(0)/a = 0 and /a e L1(T3) \ L2(T3), where the function e C(T3) is a unique (up to a constant factor) solution of Ga^a = satisfying the condition

¥>a(0) = 0.

Proof. Let the operator ha(0) have a zero energy resonance. One can see that the function /a defined in Lemma 4.4 satisfies ha(0)/a = 0. Let us show that /a e L1(T3) \L2(T3). First we recall that the solution of Ga^a = is equal to ^a(p) = 1 (up to constant factor). The definition of the function e(-) implies the existence of positive constants C1,C2,C3 and 5 such that:

C1|q - p'|2 < e(q) < C2|q - p'|2, q e Us(p'), p' e A; (4.2)

-(q) - C3, q G T. (4.3)

Using the estimates (4.2) and (4.3) we have:

ds c f ds

- 2 J R4

|/«(s)|2 ds - / — - C J 1-14 = œ;

T3 Us(0) Us(0)

i l/a(s)|ds = £ i -dS)+ i < Ci £ i + C3 < œ.

J P^A J -(S) J -(S) J |S - P'12

T3 peA Us(p') Ts peAUs(p')

Therefore, / G Li(T3) \ L2OT3). □ We denote:

m« + mY mi + m2 + m3

m«7 :=-L, :=----.

m« m7 ma(m« + m7 )

Now, we formulate a lemma (zero energy expansion for the Fredholm determinant, leading to behaviors of the zero energy resonance), which is important in the proof of Theorem 3.3, that is, the asymptotics (3.2).

Lemma 4.5. Let pa = p°a and g A. Then, the following decomposition:

,0 3/2

uuramJY 2z

Aa(K - p ; z - £a(p)) = V UalP - P|2 - ^ + ^ - P'^ + O(|z|)

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

Proof. Let us sketch the main idea of the proof. Take a sufficiently small 5 > 0 such that Us(p') n Us(q') = 0 for all q' e A with q' = p'. Let pa = p°a and K,p' e A. Using the additivity of the integral, we rewrite the function Aa(K - p; z - ea(p)) as:

Aa(K - p ; z - £a(p)) =

,'0

1 -

(

Va

(2n)3

ds ds

+

q'eArr/n Eia)(s) + -a(P) - z / Eia)(s) + -a(P) - z Us (q') Ts /

(4.4)

Since the function (■, ■) has non-degenerate zero minima at the points (p',q'), p',q' G A, analysis similar to [17] shows that:

f ds f ds

J Eia)(-s}+ e«(p) - z J £«(p')

Us (g') Us (g')

4n2mi{2 I 2z , o

-2T^\ na|p - p'|2 - -2 + O(|p - p'|2) + O(|z|);

n2 V n2

i -n-ds-= i ( ) ds-+ O(|p - p'|2) + O(|z|)

EPa)(s) + £«(p) - z J )(s) + Up') (|P P | ) (| |)

as |p - p'| ^ 0 and z ^ -0. Substituting the last two expressions into (4.4), we obtain:

0 3/2 I---

A«(K - p; z - e«(p)) = A«(K - p'; 0) + /nQ|p - p'|2 - -2 + O(|p - p'|2) + O(|z|)

2n V n2

as |p-p'| ^ 0 and z ^ -0. Now, the equality = ^i, that is, Aa(K-p'; 0) = 0 completes the proof of Lemma 4.5. □

Corollary 4.6. Let = ^ and K g A. For some CI, C2, C3 > 0 and £ > 0 the following inequalities hold:

(i) Ci|p - p'| < A«(K - p; -e«(p)) < C2|p - p'|, p G U(p'), p' G A;

(ii) Aa(K - p; -e«(p)) > C3, p G T.

Proof. Lemma 4.5 yields the assertion (i) for some positive numbers C1,C2. The positivity

and continuity of the function Aa(-; -£«(•)) on the compact set T imply the assertion (ii).

□

5. The Birman-Schwinger principle

For a bounded self-adjoint operator A acting in the Hilbert space R, we define the number n(Y, A) as follows:

n(Y, A) = sup{dimF : (Au,u) > 7, u G F cR, ||u|| = 1}.

The number n(Y, A) is equal to infinity if 7 < maxaess(A); if n(Y, A) is finite, then it is equal to the number of the eigenvalues of A larger than 7. By the definition of N(K; z), we have:

N(K; z) = n(-z, -H(K)), -z > -Tess(K).

Let > 0 and K g T3. Then we have Aa(K - p ; z - e«(p)) > 0 for any p G T3 and

z < Tess(K).

In what follows, we deal with the operators in various spaces of vector-valued functions. They will be denoted by bold letters and will be written in matrix form.

Let Q c R3 be the measurable set and L2m)(Q) be the Hilbert space of m-component vector functions w = (wi,..., wm), w, G L2(Q), i = 1,... ,m.

In our analysis of the discrete spectrum of H(K), the crucial role is played by the 3 x 3 self-adjoint block operator matrix T(K; z), z < Tess(K) acting on l23)(T3) with the

entries Tap(K; z), where (K; z), a < fl is the integral operator on L2(T3) with kernel

Taa (K; z; ', ') •

Taa(K; z; p, q) = 0;

- (Ek(p, q) - z)-1

T12(K; z; p,q)

Ti3(K; z; p,q)

T23(K; z; p,q)

\JAi(K - p; z - ei(p)) ^(K - q ; z - ^(q))

___(Ek(p, q - p) - z)-1

VAi(K - p; z - ei(p)) ^(q ; z - e3(K - q))

___(Ek(q - p,p) - z

V^A2(K - p; z - e2(p)^Aa(q ; z - £3(K - q))'

and for a > ^ the operator TO,(K; z) is the adjoint operator to T^K; z).

The following lemma is a realization of the well-known Birman-Schwinger principle for three-particle Schrodinger operators on a lattice (see [7,16,17]).

Lemma 5.1. For z < Tess(K) the operator T(K; z) is compact and continuous in z and

N(K; z) = n(1, T(K; z)).

For the proof of the lemma, we refer to [17].

6. Asymptotics for the number of negative eigenvalues of H(0)

In this section, we derive the asymptotic relation (3.2) for the number of negative eigenvalues of H(0).

First, we recall that T(0; z) = T(K; z) for all K G A. Let S2 be the unit sphere in R3 and a = L2(S2). As we shall see, the discrete spectrum asymptotics of the operator T(0; z) as z ^ -0 is determined by the integral operator Sr, r = 1/21 log |z|| in L2((0, r), a(3)) with the kernel S«,(y, t), y = x - x', x, x' G (0, r), t = (£, n), £, n G S2, where:

Saa(y,t) = 0; S«,(y, t) - 1 '

4n2 cosh(y + r«,) + s«,t' / _ 1 _ (m<*7)i/2

u«^ = ka,—2- , = — log—1, ,

nan^ J 2 ma7 m7

being such that = 1 if both subsystems a and ft have zero energy resonances, otherwise, fc«, = 0. The eigenvalue asymptotics for the operator Sr have been studied in detail by Sobolev [7], by employing an argument used in the calculation of the canonical distribution of Toeplitz operators.

Let us recall some results of [7] which are important in our work.

The coefficient in asymptotics (3.2) of N(0; z) will be expressed by means of the self-adjoint integral operator S(9), 9 g R, in the space a(3), whose kernel is of the form:

1 ir e sinh[9 arccos s^t]

Saa(9,t) = 0; Sa,3 (9,t) = -¡—2ua,e

4n2 ' h „2

a,

y/1 - ^t sinh(n9)

and depends on the inner product t = (£,n) of the arguments e S2. For 7 > 0, we define:

U(7) -¿/"(Y,

This function was studied in detail in [7]; it is used in the existence proof for the Efimov effect. In particular, the function U(■) is continuous in 7 > 0, and the limit:

lim1 r-1n(Y, Sr) = U(7), (6.1)

r^o 2

exists such that U(1) > 0.

Theorem 3.3 can be derived by using a perturbation argument based on the following

lemma.

Lemma 6.1. Let A(z) = A0(z) + A1 (z), where A0(z) (A1(z)) is compact and continuous for z < 0 (for z < 0). Assume that the limit lim f (z) "(y,A0(z)) = /(7) exists and /(■)

is continuous in (0;+ro) for some function f (■), where f (z) ^ 0 as z ^ -0. Then, the same limit exists for A(z) and lim f (z) "(7, A(z)) = /(7).

For the proof of Lemma 6.1, see Lemma 4.9 in [7].

Remark 6.2. Since the function U(■) is continuous with respect to 7, it follows from Lemma 6.1 that any perturbation of A0(z) treated in Lemma 6.1 (which is compact and continuous up to z = 0) does not contribute to the asymptotic relation (3.2).

Now, we are going to reduce the study of the asymptotics for the operator T(0; z) to that of the asymptotics Sr.

Let T(£; |z|) be the 3 x 3 block operator matrix in l23)(T3) whose entries Tal3(£; |z|) are integral operators with the kernel Tap(£; |z|; ■, ■) :

T„a(i; |z|;p,q) = 0;

T«i|z|; p,q) =

Daß £

Xs(p - pOxs(q - 9')(na|p - pf + 2|z|/(n2)) 1 (nß|q - q'|2 + 2|z|/(n2)) 1

aß pw^a m-'1|p - p/|2 + 2m-1(p - p;,q- q/) + m/-1|q - q/|2 + 2|z|/(n2) '

where

-3/4 -3/4

Daß = 2n%ß7 , a,ß,Y =1, 2, 3, a = ß = 7, and x<s(■) is the characteristic function of the domain Us(0).

Lemma 6.3. Let ßa = ßß = ßß, ßY < ß0. any z < 0 and sufficiently small 5 > 0,

the difference T(0; z) - T(5; |z|) belongs to the Hilbert-Schmidt class, and is continuous with respect to z < 0.

Proof. We prove the lemma in the case ßa = ßa, a = 1,2,3. The case ßa = ßa, ßß = ßß, ß7 < ßY can be proven similarly.

By the definition of ea(-), we have:

n2

£a(P) = — |p - p/|2 + O(|p - p/|4),

as |p — p,| ^ 0 for p/ e A, which implies the expansion:

4*1 p(q) + e«(p) = n2

|p — p/|2 + (p — — qQ + |q — q/|2

2m,

'«7

mY

2m

ßY J

+ O(|p — p/|4) + O(|q — q/14)

as |p — p'|, |q — q'| ^ 0 for K,p', q' G A. Then, there exist C1, C2 > 0 and £ > 0 such that:

Ci(|p — p'|2 + |q — q'|2) < EK-p(q) + £a(p) < C2(|p — p'|2 + |q — q'|2), (p,q) G Us(p') x Us(q') for K,p',q' G A;

4-P(q)+ £a(p) > Ci, (p,q) G T2, K G T3.

Applying last estimates and Corollary 4.6, we obtain that there exists C1 > 0 such that the kernel of the operator T^(0; z) — Tap(£; |z|) can be estimated by the square-integrable function:

C

i

p',q'eA

- p/l-1/2

+

|p — p/1

- q/|-1/2

+

|q — q/|

|p — p/|2 + |q — q/|2 |p — p/12 + |q — q/|2 |p — p/|2 + |q — q/|2

+ 1

Hence, the operator Taß(0; z) — Taß(£; |z|) belongs to the Hilbert-Schmidt class for all z < 0. In combination with the continuity of the kernel of the operator with respect to z < 0, this implies the continuity of Taß(0; z) — Taß(£; |z|) with respect to z < 0. The lemma is proved. □

The following theorem is fundamental for the proof of the asymptotic relation (3.2). Theorem 6.4. The following relation holds

n(Y, T(i; |z|))

lim l*K0

1 log |z|

U(y), Y > 0.

(6.2)

Proof. First we prove Theorem 6.4 under the condition that all two-particle operators have zero energy resonances, that is, in the case where = ^i, a = 1,2,3. The case where only two operators ha(0) and hp(0) have zero energy resonance can be proven similarly.

The subspace of vector functions w = (w1,w2,w3) with components having support in (J Us(p') is an invariant subspace for the operator T(£; |z|).

p'eA

Let T0(#; |z|) be the restriction of the operator T($; |z |) to the subspace L>3)( U Us(p')),

p'eA

that is, a 3 x 3 block operator matrix in

l23)( U Us (p')) whose entries T^; | z|) are the

p'eA

integral operators with the kernel T^^; |z|; ■, ■), where T^« (£; |z|;p,q) = 0 and the function

Tg^; |z|; ■, ■) is defined on U Us(p/) x U Us(q/) as:

p' eA q'eA

«/3 ( ; I I;p,q) a/ m^Ip — p'|2 + 2m-1(p — p',q — q') + m^Iq — q'|2 + 2|z|/(n2), (p, q) G Us(p') x Us(q') for p',q' G A.

In the remainder of the proof, for convenience, we numerate the points of A as p1,... ,pn3 and set 1, n = 1,..., n.

(na|p — p/|2 + 2|z|/(n2))-4 (nß|q — q/|2 + 2|z|/(n2))

1

4

n3 n3

Since L2(U U(p,)) = 0 L2(Us(p,)), we can express the integral operator T^g^; |z|)

as the following n3 x n3 block operator matrix T^^; |z|) acting on 0 L2(U5(p,)) as:

i=i

^ Tiß'1)(i; |z|) ... T^i; |z|) \

TSS(i; |z|)

V T(n д)(£;|z|) ... ^n,n)(^;lz|) У

,

where T^^; |z|) • L2(U,s(pj)) ^ L2(U,s(pi)) is an integral operator with the kernel

TS^; |z|; p, q), (p, q) G Us (pi) x Us (p) for i, j = 1,n3.

It is easy to show that T0(8; |z|) is unitarily equivalent to the 3 x 3 block operator matrix T(r), r = |z|-1, acting on L^3)(Ur(0))©L^3)(Ur(0))©L^3)(Ur(0)) with the entries Ta3(r): L2n3)(Ur(0)) ^ L2n3)(Ur(0)) •

/ Ta,(r) ... Ta3(r)

Taa(r) = 0; Ta, (r) = I . ... .

\ Ta,(r) ... Ta3(r)

where Ta,(r) is the integral operator on L2(Ur(0)) with the kernel:

D

(na|p|2 + 2/(n2))-4 (nß|q|2 + 2/(n2))-4

a3 m-7i|p|2 + 2m-i(p,q) + m-7i|q|2 + 2/(n2)' The equivalence is realized by the unitary dilation (3n3 x 3n3 diagonal matrix):

3n3

Br • 0L2(Us(pi)) ^ L23n3)(Ur(0)),

i=i

Br = diag{B«,..., Brn3), Bri),..., Brn3), B(i),..., Brn3)}. Here, the operator B^ • L2(Us(pi)) ^ L2(Ur(0)), i = 1,n3 acts as:

(b«/)(p)= (r)-1 /(% + p^ .

'r\-2 „(8

r

Let us introduce the 3n3 x 1 and 1 x 3n3 block operator matrices:

E : ¿23n3)(Ur(0)) ^ ¿23)(Ur(0)), Ar : l23)(U(0)) ^ L23n3)(U(0))

of the form

i 0 Ai2(r) Ai3(r) \ Ar = I A2i(r) 0 A23(r) I , E = diag{I, I, I},

\ A3i(r) A32(r) 0 j

where Aaß(r) and I are the n3 x 1 and 1 x n3 matrices of the form:

Aa ß (r)

^aß (r)

V ^(r

I =(/.../ ),

respectively, here I is the identity operator on L2(Ur(0)).

It is well known that if B1,B2 are bounded operators and y = 0 is an eigenvalue of B1B2, then y is an eigenvalue for B2B1 as well of the same algebraic and geometric multiplicities (see e.g. [20]). Therefore, n(Y, ArE) = n(Y, EAr), y > 0. Note that Ta/(r) = Aa/(r)I and: n3Ta/(r) = IAa/(r). Hence, direct calculation shows that T(r) = ArE and

EAr : ¿23)(Ur(0)) ^ L23)(Ur(0)), EAr = n3

( 0 t11) (r) Ti(31)(r)^

T21) (r) 0 T(1)(r) V T3(1)(r) T^r) 0 )

So, n(Y, T1(r)) = n(Y, EAr), y > 0. Furthermore, we can replace:

, ,2 2 \ 1 ( . l2 2 \ 1 . |p|2 2(p, q) |q|2 2 n«|p|2 + ^ , Uß|q|2 + ^ and + + +

n2/ \ n2/ ma Y mY mßY n2

by the expressions:

(n a|p|2)4 (1 — X1(p))-1, (nß|q|2)4 (1 — X1(q))-1 and ^ + ^ + ,

m« y m7

respectively, because the corresponding difference is a Hilbert-Schmidt operator and continuous up to z = 0. In this case, we obtain the block operator matrix S(r) on ¿23)(Ur(0) \ U1(0)) whose entries Saß(r) are the integral operators with the kernel Saß(r; ■, ■) :

Saa(r;p, q) = 0; Saß(r;p, q) Using the dilation:

r;p q = n3Dß__|p|-1/21q| 1/2

aß ; , (n1n2)1/4 m-1|p|2 + 2m-1 (p, q) + 1q|2.

M = diag{M,M,M} : l23)(U(0) \ U1(0)) ^ ^((0, r),a(3)), (Mf )(x,w) = e3x/2f (exw),

where r = 21 log |z||, x G (0, r), w g S2, one can see that the operator S(r) is unitarily equivalent to the integral operator Sr.

Since the difference of the operators Sr and T(£; |z|) is compact (up to unitary equivalence) and r = 1/21 log |z||, we obtain the equality:

n n(Y,T(5; |z|)^ 1 .

lim V,,, ; = lim- r-1n(Y, Sr), y > 0.

|z|—0 I log |zII r—0 2 w' r;' '

Now Lemma 6.1 and the equality (6.1) completes the proof of Theorem 6.4. □

Proof of Theorem 3.3. Let = ^, a = 1,2,3. Using Lemmas 6.1, 6.3 and Theorem 6.4 we have:

lim t'T(z)) = U(1).

|z| —>0 I log IzII

Taking into account the last equality and Lemma 5.1, and setting U0 = U(1), we complete the proof of Theorem 3.3. □

Acknowledgements

The authors would like to thank Prof. A. Teta for helpful discussions about the results of the paper. This work was supported by the TOSCA Erasmus Mundus grant. T. H. Rasulov wishes to thank the University of L'Aquila for the invitation and hospitality.

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