# On the Discrete Spectrum of a Model Operator in Fermionic Fock Space

Authors: Zahriddin Muminov; Fudziah Ismail; Zainidin Eshkuvatov; Jamshid Rasulov
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider a model operatorH associated with a system describing three particles in interaction, without conservation of the number of particles. The operator H acts in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock spaceℱa(L2(𝕋3)) over L2(𝕋3). We admit a general form for the "kinetic" part of the Hamiltonian H, which contains a parameter γ to distinguish the two identical particles from the third one. (i) We find a critical value γ* for the parameter γ that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for γ<γ* the Efimov effect is absent, while this effect exists for any γ>γ*. (ii) In the case γ>γ* , we also establish the following asymptotics for the number N(z) of eigenvalues of H below z<Emin=infσessH:limz→EminNz/logEmin-z=𝒰0γ𝒰0γ>0, for all γ>γ*.


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## Body

## 1. Introduction

In the spectral theory of the continuous and lattice three-particle Schrödinger operators inℝ3, there is the remarkable phenomenon known asEfimov effect: if all Hamiltonians of the two-body subsystems are nonnegative and if at least two of them have a zero-energy resonance, then the three-body system has an infinite number of negative eigenvalues accumulating at zero.This remarkable spectral property was discovered by Efimov [1] and has since become the subject of many papers [2–11]. The first mathematical proof of the existence of this effect was given by Jafaev [11], and Szlachányi and Vecsernyés [8] established the asymptotics for the number of eigenvalues near the threshold of the essential spectrum.Recently, Wang [12] has proved the existence of the Efimov effect in the system with N≥4 particles in ℝ3 but in this case the properties of the spectrum have not been fully comprehended yet.In statistical physics [13, 14], solid-state physics [15, 16], and the theory of quantum fields [17–19], some important problems arise where the number of quasiparticles is bounded, but not fixed. The authers of [20] have developed geometric and commutator techniques to find the location of the spectrum and to prove absence of singular continuous spectrum for Hamiltonians without conservation of the particle number.Notice that the study of systems describingn 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 subspaceℋ(n) of the Fock space, consisting of r≤n particles [13, 16–18, 20, 21].The model operator, associated with a system describing two bosons and one particle, another nature in interaction, without conservation of the number of particles, was considered in [22, 23], and the existence of the Efimov effect was proved. This model operator acts in the direct sum of zero-, one-, and two-particle subspaces of the bosonic Fock spaceℱs(L2(𝕋3)) over L2(𝕋3).In the present paper, we consider a model operatorH, acting in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space ℱa(L2(𝕋3)) over L2(𝕋3), associated with a system describing two identical fermions and one particle, another nature in interactions, without conservation of the number of particles on the three-dimensional lattice.The main aim of the present paper is to study spectral properties for a model operatorH with emphasis on the asymptotics for the number of infinitely many eigenvalues (Efimov's effect case).We admit a general form for thekinetic part of the hamiltonian H, which contains a parameter γ to distinguish the two identical particles from the third one (this parameter would be the ratio of the mass). Under some smoothness assumptions, we obtain the following results.(i)
We find a critical valueγ* for the parameter γ that allows or forbids the Efimov effect and we prove that only for γ<γ* the Efimov effect is absent, while this effect exists for any γ>γ*.(ii)
In the caseγ>γ*, we also establish the following asymptotics for the number N(z) of eigenvalues below z<Emin:
(1)limz→Emin-N(z)|log|Emin-z||=𝒰0(γ)(𝒰0(γ)>0),∀γ>γ*.We notice that the assertion(i) is surprising and similar assertions are not true for the lattice model operators in the bosonic Fock space [22, 23], but in both cases, the presence of the Efimov effect is due to the annihilation and creation operators.However, the presence of the Efimov effect for the three-particle Schrödinger operators is due to the two-particle interaction operators (see, e.g., [8–11, 24, 25] in lattice case).We remark that for a model operatorH associated to a system describing three particles on the lattice the authors Dell'Antonio et al. [26] found an explicit value of the parameter γ, say γ*, such that only for values of γ below this number, the Efimov effect is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of the parameter γ on the symmetric sector. In case of antisymmetric wavefunction, the number γ* is a critical value for the mass ratio, where the Efimov effect is present or absent.In the continuous case, interestingly the case of a system of three fermions, two identical and different from the third one, with short-range interaction, was also considered from a more physical point of view by Petrov [27], and he also found a critical value for the mass ratio ≈13,6 (in our case γ*≈13,60696570) that permits or forbids the Efimov effect.The organization of the present paper is as follows. Section1 is an introduction to the whole work. In Section 2, the model operator is described as a bounded self-adjoint operator H in ℋ(3). Some spectral properties of the corresponding Friedrichs models h(p), p∈𝕋3, are studied, the location and structure of the essential spectrum of H are given, and the main result of the paper is formulated. Section 3 deals with the review the Birman-Schwinger principle for the operator H. In Section 4, we prove of the main result.In order to facilitate a description of the content of this paper, we briefly introduce the notation used throughout this paper. Let𝕋3=(ℝ/2πℤ)3=(-π,π]3 be the three-dimensional torus (the first Brillouin zone, i.e., the dual group of ℤ3) equipped with its Haar measure. Denote by L2(Ω) the Hilbert space of square-integrable functions defined on a measurable set Ω⊂ℝn. Denote by Las2((𝕋3)2) the subspace of antisymmetric functions of the Hilbert space L2((𝕋3)2). We denote by c, C several constants, whose magnitudes are not of interest.


## 2. The Model Operator and Statement of the Main Results

Set(2)ℋ0=ℂ,ℋ1=L2(𝕋3),ℋ2=Las2((𝕋3)2).The Hilbert spaceℋ(3)=ℋ0⊕ℋ1⊕ℋ2 is called thedirect sum of zero-, one-, and two-particle subspaces of a fermionic Fock space ℱa(L2(𝕋3)) over L2(𝕋3).LetHij be annihilation (creation) operators [18] defined in the Fock space for i<j(i>j). We note that 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.In this paper, we consider the case, where the number of annihilations and creations of the particles of the considering system is equal to 1. It means thatHij≡0 for all |i-j|>1. So, the model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, acts in the Hilbert space ℋ(3) as a matrix operator
(3)H=(H00H010H10H11H120H21H22),
where the operators Hij:ℋj→ℋi,  i,j=0,1,2 are defined by the forms
(4)(H00f0)0=u0f0,(H01f1)0=∫𝕋3‍b(q′)f1(q′)dq′,(H10f0)1=b(p)f0,(H11f1)1=u(p)f1(p),(H12f2)1=∫𝕋3‍b(q′)f2(p,q′)dq′,(H21f1)2=12(b(q)f1(p)-b(p)f1(q)),(H22f2)2=E(p,q)f2(p,q).Herefi∈ℋi, i=0,1,2, u0-fixed real number, b(·), u(·)-real-valued analytic functions on 𝕋3 and E(·,·) is a real-analytic symmetric function defined on (𝕋3)2.Under these assumptions, the operatorH is bounded and self-adjoint in ℋ(3).Throughout this paper we assume the following additional technical assumptions.Hypothesis 1.
(a) The real-analytic functionE(·,·) which is symmetric on (𝕋3)2 is even with respect to (p,q) and has a unique nondegenerate zero minimum at the origin (0,0)∈(𝕋3)2 and there exist positive definite matrix W and positive real number γ>0 such that
(5)(∂2E(0,0)∂pi∂pj)i,j=13=(1+γ)W,(6)(∂2E(0,0)∂pi∂qj)i,j=13=γW.
(b) The functions u(·) and b(·) on 𝕋3 are even and u(·) has a unique minimum at the origin.By Hypothesis1, for any p∈𝕋3 the integral
(7)∫𝕋3b2(t)dtE(p,t)-Emin
is finite and hence it defines a continuous even function on 𝕋3, which will be denoted by Λ(p).Since the functionE(p,q) has a unique nondegenerate minimum at the point (0,0)∈(𝕋3)2 and E(0,0)=Emin the function Λ(p) is positive.Hypothesis 2.
For any nonzerop∈𝕋3, the inequality Λ(p)<Λ(0) holds.Remark 1.
For the functions(8)u(p)=ε(p)+c,b(p)=ε(p),E(p,q)=γε(p+q)+ε(p)+ε(q),p,q∈𝕋3,
where c>0 is a real number and ε(·) is real valued analytic conditionally negative definite function with a unique minimum at the origin, Hypotheses 1 and 2 are fulfilled (see [22, Lemma A1]).Recall that a complex-valued bounded functionε:𝕋m→ℂ is called conditionally negative definite if ε(p)=ε(-p)¯ and
(9)∑i,j=1n‍ε(pi-pj)ziz-j≤0
for all p1,…,pn∈𝕋m and all z=(z1,…,zn)∈ℂn satisfying ∑i=1n‍zi=0.


### 2.1. The Friedrichs Model

To formulate the main results of the paper we introduce a family of Friedrichs modelsh(p), p∈𝕋3, which act in ℋ(2)≡ℋ0⊕ℋ1 with the entries
(10)(h00(p)f0)0=u(p)f0,h01=12H01,h10=h01*,(h11(p)f1)1(q)=𝔢p(q)f1(q),
where 𝔢p(·)=E(p,·).Let the operatorh0(p), p∈𝕋3, act in ℋ(2) as
(11)h0(p)(f0f1(q))=(0𝔢p(q)f1(q)).The perturbationh(p)-h0(p) of the operator h0(p) is a self-adjoint operator of rank 2. Therefore, in accordance with the invariance of the essential spectrum under finite rank perturbations, the essential spectrum σess(h(p)) of h(p) fills the following interval on the real axis:
(12)σess(h(p))=[m(p),M(p)],
where
(13)m(p)=minq∈𝕋3𝔢p(q),M(p)=maxq∈𝕋3𝔢p(q).Definition 2.
Letu(0)≠Emin. The operator h(0) is said to have a threshold resonance if the number 1 is an eigenvalue of the operator
(14)(Gψ)(q)=b(q)2(u(0)-Emin)∫𝕋3‍b(t)ψ(t)dt𝔢0(t)-Emin,ψ∈C(𝕋3),
and the associated eigenfunction ψ (up to a constant factor) satisfies the condition ψ(0)≠0.Remark 3.
The spectrum and resonances of this Friedrichs model are studied in [22, 23].For anyp∈𝕋3, we define an analytic function Δ(p,z) (the Fredholm determinant associated with the operator h(p)) in ℂ∖[m(p),M(p)] by
(15)Δ(p,z)=u(p)-z-12∫𝕋3‍b2(t)dtE(p,t)-z.
Since Δ(0,·) is continuous in z≤Emin, the following finite limit exists:
(16)Δ(0,Emin)=limz→EminΔ(0,z).Lemma 4.
For anyp∈𝕋3 the operator h(p) has an eigenvalue z∈ℂ∖[m(p),M(p)] if and only if Δ(p,z)=0.Proof.
One can see that the equation(17)h(p)f=zf,f∈ℋ(2),z∈ℂ∖[m(p),M(p)],
is equivalent to the following equation:
(18)(h00(p)-z)f0+h01f1=0,f1=-(h11(p)-z)-1h10f0.
Puttingf1 to the first equation and then according to the equality Δ(p,z)=h00(p)-z-h01(h11(p)-z)-1h10 we have the fact that (17) is equivalent to the equation
(19)Δ(p,z)f0=0.
Thus (17) has nontrivial solution if and only if Δ(p,z)=0.Lemma 5.
(i) The operatorh(0) has a threshold resonance if and only if Δ(0,Emin)=0 and b(0)≠0.(ii)
Ifu(0)≤Emin, then the operator h(0) has no a threshold resonance.(iii)
AssumeΔ(0,Emin)=0.
(a)
Ifb(0)≠0, then the operator h(0) has a threshold resonance and the vector f=(f0,f1), where
(20)f0=const≠0,f1(q)=-b(q)f02(𝔢0(q)-Emin)∈L1(𝕋3)∖L2(𝕋3),
obeys the equation h(0)f=Eminf.(b)
Ifb(0)=0, then the number z=Emin is an eigenvalue of the operator h(0) and the vector f=(f0,f1), where f0∈ℂ1 and f1∈L2(𝕋3) defined by (20), is the corresponding eigenvector.Proof.
(i) “Only If Part.” Suppose that the operatorh(0) has a threshold energy resonance. Then by Definition 2 the number 1 is an eigenvalue of the operator (14) as
(21)(Gψ)(q)=b(q)2(u(0)-Emin)∫𝕋3‍b(t)ψ(t)dt𝔢0(t)-Emin,iψ∈C(𝕋3),
and the associated eigenfunction ψ satisfies ψ(0)≠0.
This solution is equal to the functionb(·) (up to a constant factor) and hence Δ(0,Emin)=0.
“If Part.” Let the equalityΔ(0,Emin)=0 hold and let b(0)≠0. Then the inequality u(0)≠Emin holds and the function b(·) is an eigenfunction of G; that is, by Definition 2 the operator h(0) has a threshold energy resonance.(ii)
Sinceu(0)≤Emin and -h01(h11(0)-Emin)-1h10<0, we get Δ(0,Emin)<0 which ends the proof together with (ii).(iii)
Similarly to the proof of Lemma4, one can check that Δ(0,Emin)=h00(0)-Emin-h01(h11(0)-Emin)-1h10 holds, and it is a determinant of the equation
(22)(h00(p)-Emin)f0+h01f1=0,f1=-(h11(0)-Emin)-1h10f0,thatisf1(q)=-b(q)f02(𝔢0(q)-Emin)-1,which is equivalent to the equation
(23)h(0)f=Eminf,f∈ℂ⊕L1(𝕋3).
Thus (23) has nontrivial solution if and only if Δ(0,Emin)=0.
By Hypothesis1 there exist c, C>0 and a bounded, partially continuous b¯(·) function such that
(24)c|q|2≤𝔢0(q)-Emin≤C|q|2,b(q)=b(0)+b¯(q)|q|2.(a)
Ifb(0)≠0 by (i), the operator h(0) has a threshold resonance. Due to the fact 1/|q|2∈L1(𝕋3)∖L2(𝕋3) and (24), we have f1∈L1(𝕋3)∖L2(𝕋3).(b)
Ifb(0)=0, then using (24) we can see f1∈L2(𝕋3) and f=(f0,f1)∈ℋ(2), with f eigenfunction of h(0) corresponding to the eigenvalue Emin.LetW be the 3×3 matrix defined in Hypothesis 1 and Uδ(0)={p∈𝕋3:|p|<δ} stands for a δ(δ>0)-neighborhood of the origin. The following Lemma 6 plays a crucial role in the proof of the infiniteness (finiteness, resp.) of the number of eigenvalues lying below the bottom of the essential spectrum for a model operator H. The proof may be handled in much the same way as in [22, 23].Lemma 6.
Let the operatorh(0) have a threshold resonance. Then for any p∈Uδ(0), δ>0 sufficiently small and z≤Emin, the following decomposition
(25)Δ(p,z)=42π2b2(0)(1+γ)3/2det(W)1/2m(p)-z+Δ(02)(m(p)-z)+Δ(20)(p,z)
holds, where Δ(02)(m(p)-z) (Δ(20)(p,z), resp.) is a function behaving like O(m(p)-z) (O(|p|2), resp.) as |m(p)-z|→0 (p→0 uniformly in z≤Emin, resp.).Lemma 7.
Let the operatorh(0) have a threshold resonance. Then there exist positive numbers c, C and δ such that
(26)c|p|≤Δ(p,Emin)≤C|p|foranyp∈Uδ(0),(27)Δ(p,Emin)≥cforanyp∈𝕋3∖Uδ(0).Proof.
By the assumption (a) of Hypothesis1, we have
(28)E(p,q)=Emin+F(p,q)+O(|p|4+|q|4)asp,q→0,
where
(29)F(p,q)=12((1+γ)(Wp,p)+2γ(Wp,q)+(1+γ)(Wq,q)).
SinceE(·,·) has a nondegenerate minimum at the origin and the equality
(30)minq∈𝕋3F(p,q)=121+2γ1+γ(Wp,p)
Holds, we have
(31)m(p)=minq∈𝕋3E(p,q)=Emin+121+2γ1+γ(Wp,p)+O(|p|4)asp→0.
Consequently, these asymptotics and Lemma6 yield (26) for some positive numbers c, C.
By Hypotheses1 and 2, we have u(p)>u(0) and Λ(p)>Λ(0), 0≠p∈𝕋3, and hence
(32)Δ(p,Emin)=u(p)-Emin-12Λ(p)>Δ(0,Emin),0≠p∈𝕋3.
Then the inequality (27) follows from Δ(0,Emin)=0 and the continuity of the function Δ(p,Emin).


### 2.2. The Essential Spectrum ofH

The following theorem describes the essential spectrum of the operatorH. Similar results were obtained in the bosonic Fock space in [22, 23], and we refer to these paper for the proof.Theorem 8.
For the essential spectrumσ<?brm?>ess<?erm?>(H) of the operator H, the equality
(33)σ<?brm?>ess<?erm?>(H)=⋃p∈𝕋3σd(h(p))∪[Emin,Emax]
holds, where σd(h(p)) is the discrete spectrum of the operator h(p), p∈𝕋3.The following lemma describes the location of the essential spectrum of the operatorH.Lemma 9.
Let Hypothesis1 be fulfilled and h(0) has a threshold resonance. Then
(34)σ<?brm?>ess<?erm?>(H)=[Emin,Emax]∪σ2,
where σ2={z∈[Emax,+∞):Δ(p,z)=0forsomep∈𝕋3}.Proof.
According to Lemma4, the zero of Δ(p,·) is the eigenvalues of h(p), so
(35)ϒ=⋃p∈𝕋3σd(h(p))∖(Emin,Emax)={z∈ℝ∖[Emin,Emax]:Δ(p,z)=0forsomep∈𝕋3}.
By analyticity ofΔ(p,·), the subset ϒ of the essential spectrum of σ<?brm?>ess<?erm?>(H) is described as a finite union of nonintersecting closed intervals.
The functionΔ(p,·) is decreasing in (-∞,Emin), respectively, with (Emax,+∞), and hence there may exist its unique zero lying below, respectively, above the closed interval [Emin,Emax].
So, the setϒ may be described as the union of intervals σ1 and σ2 lying below and above of [Emin,Emax], and
(36)σ1={z∈(-∞,Emin]:Δ(p,z)=0forsomep∈𝕋3},σ2={z∈[Emax,+∞):Δ(p,z)=0forsomep∈𝕋3}.
By virtue of Hypothesis2, we have Δ(p,Emin)≥Δ(0,Emin) and using Lemma 5 we get Δ(0,Emin)=0, and so Δ(p,z)>Δ(p,Emin)≥Δ(p,Emin)=0, that is
(37)Δ(p,z)>0∀z<Emin,
since Δ(p,·) is decreasing in (-∞,Emin). Then according to Lemma 4 we get σ1=∅.
Consequently,∪p∈𝕋3σd(h(p))∖(Emin,Emax)=σ2 and Theorem 8 ends the proof of the lemma.


### 2.3. Statement of the Main Results

Henceforth we assume thath(0) has a threshold resonance.According to Lemma9 for the bottom of the essential spectrum the equality Emin=infσ<?brm?>ess<?erm?>(H) is valid, and by N(z) we denote the number of eigenvalues and counted according to their multiplicities of H lying below z≤Emin.Letγ* be a solution of the equation
(38)2(1+γ)2πγ1+2γ-2(1+γ)2πγ2arcsinγ1+γ=1,γ>0.Since the function at the r.h.s. of (38) is continuous, strictly increasing and surjective from (0,∞) to (0,∞), the number γ* is a unique positive solution.In the following main theorem, we aprecisely describe the dependence of the number of eigenvalues ofH on the parameters γ>0.Theorem 10.
Let Hypothesis1 be fulfilled and let h(0) have a threshold resonance at Emin. Then we have the following. (i)
For any0<γ<γ*, the operator H has a finite number of eigenvalues lying below the bottom Emin of the essential spectrum.(ii)
For anyγ>γ*, the operator H has infinitely many eigenvalues lying below Emin. The function N(z) obeys the relation
(39)limz→Emin-N(z)|log|Emin-z||=𝒰0(γ)(𝒰0(γ)>0).It follows from the positivity of the limit (39) that the discrete spectrum of the operator H is infinite.Remark 11.
The constant𝒰0(γ) is given as a positive function depending only on the variable γ, γ>γ*.Remark 12.
In [26], a result analogue to Theorem 10 has been proven for a model operator H associated to a system describing three particles on the lattice ℤd.


## 2.1. The Friedrichs Model

To formulate the main results of the paper we introduce a family of Friedrichs modelsh(p), p∈𝕋3, which act in ℋ(2)≡ℋ0⊕ℋ1 with the entries
(10)(h00(p)f0)0=u(p)f0,h01=12H01,h10=h01*,(h11(p)f1)1(q)=𝔢p(q)f1(q),
where 𝔢p(·)=E(p,·).Let the operatorh0(p), p∈𝕋3, act in ℋ(2) as
(11)h0(p)(f0f1(q))=(0𝔢p(q)f1(q)).The perturbationh(p)-h0(p) of the operator h0(p) is a self-adjoint operator of rank 2. Therefore, in accordance with the invariance of the essential spectrum under finite rank perturbations, the essential spectrum σess(h(p)) of h(p) fills the following interval on the real axis:
(12)σess(h(p))=[m(p),M(p)],
where
(13)m(p)=minq∈𝕋3𝔢p(q),M(p)=maxq∈𝕋3𝔢p(q).Definition 2.
Letu(0)≠Emin. The operator h(0) is said to have a threshold resonance if the number 1 is an eigenvalue of the operator
(14)(Gψ)(q)=b(q)2(u(0)-Emin)∫𝕋3‍b(t)ψ(t)dt𝔢0(t)-Emin,ψ∈C(𝕋3),
and the associated eigenfunction ψ (up to a constant factor) satisfies the condition ψ(0)≠0.Remark 3.
The spectrum and resonances of this Friedrichs model are studied in [22, 23].For anyp∈𝕋3, we define an analytic function Δ(p,z) (the Fredholm determinant associated with the operator h(p)) in ℂ∖[m(p),M(p)] by
(15)Δ(p,z)=u(p)-z-12∫𝕋3‍b2(t)dtE(p,t)-z.
Since Δ(0,·) is continuous in z≤Emin, the following finite limit exists:
(16)Δ(0,Emin)=limz→EminΔ(0,z).Lemma 4.
For anyp∈𝕋3 the operator h(p) has an eigenvalue z∈ℂ∖[m(p),M(p)] if and only if Δ(p,z)=0.Proof.
One can see that the equation(17)h(p)f=zf,f∈ℋ(2),z∈ℂ∖[m(p),M(p)],
is equivalent to the following equation:
(18)(h00(p)-z)f0+h01f1=0,f1=-(h11(p)-z)-1h10f0.
Puttingf1 to the first equation and then according to the equality Δ(p,z)=h00(p)-z-h01(h11(p)-z)-1h10 we have the fact that (17) is equivalent to the equation
(19)Δ(p,z)f0=0.
Thus (17) has nontrivial solution if and only if Δ(p,z)=0.Lemma 5.
(i) The operatorh(0) has a threshold resonance if and only if Δ(0,Emin)=0 and b(0)≠0.(ii)
Ifu(0)≤Emin, then the operator h(0) has no a threshold resonance.(iii)
AssumeΔ(0,Emin)=0.
(a)
Ifb(0)≠0, then the operator h(0) has a threshold resonance and the vector f=(f0,f1), where
(20)f0=const≠0,f1(q)=-b(q)f02(𝔢0(q)-Emin)∈L1(𝕋3)∖L2(𝕋3),
obeys the equation h(0)f=Eminf.(b)
Ifb(0)=0, then the number z=Emin is an eigenvalue of the operator h(0) and the vector f=(f0,f1), where f0∈ℂ1 and f1∈L2(𝕋3) defined by (20), is the corresponding eigenvector.Proof.
(i) “Only If Part.” Suppose that the operatorh(0) has a threshold energy resonance. Then by Definition 2 the number 1 is an eigenvalue of the operator (14) as
(21)(Gψ)(q)=b(q)2(u(0)-Emin)∫𝕋3‍b(t)ψ(t)dt𝔢0(t)-Emin,iψ∈C(𝕋3),
and the associated eigenfunction ψ satisfies ψ(0)≠0.
This solution is equal to the functionb(·) (up to a constant factor) and hence Δ(0,Emin)=0.
“If Part.” Let the equalityΔ(0,Emin)=0 hold and let b(0)≠0. Then the inequality u(0)≠Emin holds and the function b(·) is an eigenfunction of G; that is, by Definition 2 the operator h(0) has a threshold energy resonance.(ii)
Sinceu(0)≤Emin and -h01(h11(0)-Emin)-1h10<0, we get Δ(0,Emin)<0 which ends the proof together with (ii).(iii)
Similarly to the proof of Lemma4, one can check that Δ(0,Emin)=h00(0)-Emin-h01(h11(0)-Emin)-1h10 holds, and it is a determinant of the equation
(22)(h00(p)-Emin)f0+h01f1=0,f1=-(h11(0)-Emin)-1h10f0,thatisf1(q)=-b(q)f02(𝔢0(q)-Emin)-1,which is equivalent to the equation
(23)h(0)f=Eminf,f∈ℂ⊕L1(𝕋3).
Thus (23) has nontrivial solution if and only if Δ(0,Emin)=0.
By Hypothesis1 there exist c, C>0 and a bounded, partially continuous b¯(·) function such that
(24)c|q|2≤𝔢0(q)-Emin≤C|q|2,b(q)=b(0)+b¯(q)|q|2.(a)
Ifb(0)≠0 by (i), the operator h(0) has a threshold resonance. Due to the fact 1/|q|2∈L1(𝕋3)∖L2(𝕋3) and (24), we have f1∈L1(𝕋3)∖L2(𝕋3).(b)
Ifb(0)=0, then using (24) we can see f1∈L2(𝕋3) and f=(f0,f1)∈ℋ(2), with f eigenfunction of h(0) corresponding to the eigenvalue Emin.LetW be the 3×3 matrix defined in Hypothesis 1 and Uδ(0)={p∈𝕋3:|p|<δ} stands for a δ(δ>0)-neighborhood of the origin. The following Lemma 6 plays a crucial role in the proof of the infiniteness (finiteness, resp.) of the number of eigenvalues lying below the bottom of the essential spectrum for a model operator H. The proof may be handled in much the same way as in [22, 23].Lemma 6.
Let the operatorh(0) have a threshold resonance. Then for any p∈Uδ(0), δ>0 sufficiently small and z≤Emin, the following decomposition
(25)Δ(p,z)=42π2b2(0)(1+γ)3/2det(W)1/2m(p)-z+Δ(02)(m(p)-z)+Δ(20)(p,z)
holds, where Δ(02)(m(p)-z) (Δ(20)(p,z), resp.) is a function behaving like O(m(p)-z) (O(|p|2), resp.) as |m(p)-z|→0 (p→0 uniformly in z≤Emin, resp.).Lemma 7.
Let the operatorh(0) have a threshold resonance. Then there exist positive numbers c, C and δ such that
(26)c|p|≤Δ(p,Emin)≤C|p|foranyp∈Uδ(0),(27)Δ(p,Emin)≥cforanyp∈𝕋3∖Uδ(0).Proof.
By the assumption (a) of Hypothesis1, we have
(28)E(p,q)=Emin+F(p,q)+O(|p|4+|q|4)asp,q→0,
where
(29)F(p,q)=12((1+γ)(Wp,p)+2γ(Wp,q)+(1+γ)(Wq,q)).
SinceE(·,·) has a nondegenerate minimum at the origin and the equality
(30)minq∈𝕋3F(p,q)=121+2γ1+γ(Wp,p)
Holds, we have
(31)m(p)=minq∈𝕋3E(p,q)=Emin+121+2γ1+γ(Wp,p)+O(|p|4)asp→0.
Consequently, these asymptotics and Lemma6 yield (26) for some positive numbers c, C.
By Hypotheses1 and 2, we have u(p)>u(0) and Λ(p)>Λ(0), 0≠p∈𝕋3, and hence
(32)Δ(p,Emin)=u(p)-Emin-12Λ(p)>Δ(0,Emin),0≠p∈𝕋3.
Then the inequality (27) follows from Δ(0,Emin)=0 and the continuity of the function Δ(p,Emin).


## 2.2. The Essential Spectrum ofH

The following theorem describes the essential spectrum of the operatorH. Similar results were obtained in the bosonic Fock space in [22, 23], and we refer to these paper for the proof.Theorem 8.
For the essential spectrumσ<?brm?>ess<?erm?>(H) of the operator H, the equality
(33)σ<?brm?>ess<?erm?>(H)=⋃p∈𝕋3σd(h(p))∪[Emin,Emax]
holds, where σd(h(p)) is the discrete spectrum of the operator h(p), p∈𝕋3.The following lemma describes the location of the essential spectrum of the operatorH.Lemma 9.
Let Hypothesis1 be fulfilled and h(0) has a threshold resonance. Then
(34)σ<?brm?>ess<?erm?>(H)=[Emin,Emax]∪σ2,
where σ2={z∈[Emax,+∞):Δ(p,z)=0forsomep∈𝕋3}.Proof.
According to Lemma4, the zero of Δ(p,·) is the eigenvalues of h(p), so
(35)ϒ=⋃p∈𝕋3σd(h(p))∖(Emin,Emax)={z∈ℝ∖[Emin,Emax]:Δ(p,z)=0forsomep∈𝕋3}.
By analyticity ofΔ(p,·), the subset ϒ of the essential spectrum of σ<?brm?>ess<?erm?>(H) is described as a finite union of nonintersecting closed intervals.
The functionΔ(p,·) is decreasing in (-∞,Emin), respectively, with (Emax,+∞), and hence there may exist its unique zero lying below, respectively, above the closed interval [Emin,Emax].
So, the setϒ may be described as the union of intervals σ1 and σ2 lying below and above of [Emin,Emax], and
(36)σ1={z∈(-∞,Emin]:Δ(p,z)=0forsomep∈𝕋3},σ2={z∈[Emax,+∞):Δ(p,z)=0forsomep∈𝕋3}.
By virtue of Hypothesis2, we have Δ(p,Emin)≥Δ(0,Emin) and using Lemma 5 we get Δ(0,Emin)=0, and so Δ(p,z)>Δ(p,Emin)≥Δ(p,Emin)=0, that is
(37)Δ(p,z)>0∀z<Emin,
since Δ(p,·) is decreasing in (-∞,Emin). Then according to Lemma 4 we get σ1=∅.
Consequently,∪p∈𝕋3σd(h(p))∖(Emin,Emax)=σ2 and Theorem 8 ends the proof of the lemma.


## 2.3. Statement of the Main Results

Henceforth we assume thath(0) has a threshold resonance.According to Lemma9 for the bottom of the essential spectrum the equality Emin=infσ<?brm?>ess<?erm?>(H) is valid, and by N(z) we denote the number of eigenvalues and counted according to their multiplicities of H lying below z≤Emin.Letγ* be a solution of the equation
(38)2(1+γ)2πγ1+2γ-2(1+γ)2πγ2arcsinγ1+γ=1,γ>0.Since the function at the r.h.s. of (38) is continuous, strictly increasing and surjective from (0,∞) to (0,∞), the number γ* is a unique positive solution.In the following main theorem, we aprecisely describe the dependence of the number of eigenvalues ofH on the parameters γ>0.Theorem 10.
Let Hypothesis1 be fulfilled and let h(0) have a threshold resonance at Emin. Then we have the following. (i)
For any0<γ<γ*, the operator H has a finite number of eigenvalues lying below the bottom Emin of the essential spectrum.(ii)
For anyγ>γ*, the operator H has infinitely many eigenvalues lying below Emin. The function N(z) obeys the relation
(39)limz→Emin-N(z)|log|Emin-z||=𝒰0(γ)(𝒰0(γ)>0).It follows from the positivity of the limit (39) that the discrete spectrum of the operator H is infinite.Remark 11.
The constant𝒰0(γ) is given as a positive function depending only on the variable γ, γ>γ*.Remark 12.
In [26], a result analogue to Theorem 10 has been proven for a model operator H associated to a system describing three particles on the lattice ℤd.


## 3. The Birman-Schwinger Principle

For a bounded self-adjoint operatorB, we define n(λ,B) by
(40)n(λ,B)=sup{dimF:(Bu,u)>λ,u∈F,||u||=1}.n(λ,B) is equal to infinity if λ is in the essential spectrum of B and if n(λ,B) is finite, it is equal to the number of the eigenvalues of B larger than λ. By the definition of N(z), we have
(41)N(z)=n(-z,-H),-z>-Emin.In our analysis of the spectrum ofH, the crucial role is played by the self-adjoint compact Faddeev-Newton-type integral operator T(z), z<Emin, in the space ℋ(2) with the entries
(42)(T00(z)f0)0=(1-u0-z)f0,(T01(z)f1)0=-∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,z),T10(z)=T01*(z),(T11(z)f1)1(p)=-b(p)2Δ(p,z)∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,z)(E(p,q′)-z).Now we describeT(z) with the entries of H in (3). We split H21 into the sum of two operators
(43)H21=12(H21(1)-H21(2)),(H21(1)f1)2=b(q)f1(p),(H21(2)f1)2=b(p)f1(q),
where f1∈ℋ1.According to the last representation and (3), we take
(44)T00(z)=(1+z)I0-H00,T01(z)=-H01D-1/2(z),T10(z)=-D-1/2(z)H10,T11(z)=-12D-1/2(z)H12R221/2(z)H21(2)D-1/2(z),
where R22(z)=(H22-zI2)-1 is the resolvent of H22,
(45)D(z)=H11-zI1-12D-1/2(z)H12R221/2(z)H21(1)D-1/2(z).One can verify thatD(z) is the multiplication operator by the determinant Δ(·,z). So, logically we may denote by D-1/2(z) the multiplication operator by Δ-1/2(·,z).The following lemma is a modification of the Birman-Schwinger principle for the three-particle Schrödinger operators (see [8, 10, 24]).Lemma 13.
The operatorT(z) is compact and continuous in z<Emin and
(46)N(z)=n(1,T(z)).Proof.
Set(47)M(z)=ℛ1/2(z)(ℛ-1(z)-(H-zI))ℛ1/2(z),
where ℛ(z)=diag{I0,I1,ℛ22(z)} and ℛ-1(z) is the inverse of ℛ(z).
For entries of matrix operatorM(z), we have
(48)M00(z)=(1+z)I0-H00,M01(z)=-H01,M02(z)=0,M10(z)=-H10,M11(z)=(1+z)I1-H11,M12(z)=-H12R221/2(z),M20(z)=0,M21(z)=-R221/2(z)H21,M22(z)=0.
The correctness of(49)N(z)=n(1,M(z))
is shown by the fact g∈ℋ-H(-z), that is, (Hg,g)<z(g,g) if and only if (g^,g^)<(M(z)g^,g^), g^∈ℋM(z)(1), where g^=ℛ-1/2(z)g.
For anyz<τ<?brm?>ess<?erm?>(H), we define an operator V(z) acting in the Hilbert space ℋ0⊕ℋ1 by the form
(50)V(z)=(M00(z)M01(z)M10(z)M11(z)+M12(z)M21(z)),
and prove the equality
(51)N(z)=n(1,V(z)),z<τ<?brm?>ess<?erm?>(H).
Letf=(f0,f1)∈ℋV(z)(1), and then for g=(f0,f1,-M21(z)f1)∈ℋ(3), we get
(52)(M(z)g,g)=(V(z)f,f)+(M21(z)f1,g2)>(f,f)+(g2,g2)=(g,g),
that is, g∈ℋM(z)(1).
Therefore,(53)n(1,V(z))≤n(1,M(z)),z<τ<?brm?>ess<?erm?>(H).
Using the equality(54)(V(z)f,f)=(M(z)g,g)+(M12(z)M21(z)g1,g1)-(M21(z)g1,g2)-(M12(z)g2,g1),
where g∈ℋ(3), f=(g1,g2)∈ℋ(2), and taking g=(g0,g1,g2)∈ℋM(z)(1) realized analogously, we get
(55)(V(z)f,f)>(f,f)+∥g2-M21(z)g1∥2≥(f,f),iiiiithatis,f∈ℋV(z)(1)
and so
(56)n(1,V(z))≥n(1,M(z)),z<τ<?brm?>ess<?erm?>(H).
Observe that(57)V(z)-I=𝒟1/2(z)(T(z)-I)𝒟1/2(z),
where 𝒟1/2(z) is acting in the Hilbert space ℋ0⊕ℋ1 by the equality
(58)𝒟1/2(z)=diag{I0,𝒟1/2(z)}.
Employing (57), we show g∈ℋV(z)(1), that is, (V(z)g,g)>(g,g) if and only if (T(z)f,f)>(f,f), f=𝒟-1/2(z)g∈ℋT(z)(1).
Consequently, for anyz<τ<?brm?>ess<?erm?>(H), we have
(59)n(1,V(z))=n(1,T(z)),thatisN(z)=n(1,T(z)).


### 3.1. The Birman-Schwinger Principle at the Threshold

It should be noted that the operatorT(z) can be defined as a bounded operator even for the point z=Emin by
(60)(T00(Emin)f0)0=(1-u0-Emin)f0,(T01(Emin)f1)0=-∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,z),T10(Emin)=T01*(Emin),(T11(Emin)f1)1(p)=-b(p)2Δ(p,Emin)∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,Emin)(E(p,q′)-Emin).Remark 14.
The operatorT(z) converges strongly (but not uniformly) as z→Emin-0 to T(Emin). Here we do not give the proof of this convergence. The convergence of the these types of operators was shown in [11, 25].The next lemma can be proved analogously as Lemma13.Lemma 15.
For anyz, z≤Emin, the inequality
(61)N(z)≤n(1,T(Emin))
occurs.


## 3.1. The Birman-Schwinger Principle at the Threshold

It should be noted that the operatorT(z) can be defined as a bounded operator even for the point z=Emin by
(60)(T00(Emin)f0)0=(1-u0-Emin)f0,(T01(Emin)f1)0=-∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,z),T10(Emin)=T01*(Emin),(T11(Emin)f1)1(p)=-b(p)2Δ(p,Emin)∫𝕋3‍b(q′)f1(q′)dq′Δ(q′,Emin)(E(p,q′)-Emin).Remark 14.
The operatorT(z) converges strongly (but not uniformly) as z→Emin-0 to T(Emin). Here we do not give the proof of this convergence. The convergence of the these types of operators was shown in [11, 25].The next lemma can be proved analogously as Lemma13.Lemma 15.
For anyz, z≤Emin, the inequality
(61)N(z)≤n(1,T(Emin))
occurs.


## 4. The Sketch of the Proof of the Main Results

By Hypothesis1, we get
(62)E(p,q)=Emin+12((1+γ)(Wp,p)+2γ(Wp,q)+(1+γ)(Wq,q))+O(|p|4+|q|4)
as p,q→0 and (31) as
(63)m(k)=Emin+121+2γ1+γ(Wk,k)+O(|k|4)ask→0.Applying the asymptotics form(p) and using Lemma 6, we have
(64)Δ(p,z)=4π2b2(0)(1+γ)3/2det(W)1/2[n(Wp,p)+2(Emin-z)]1/2+O(|p|2+|Emin-z|)asp,|Emin-z|→0,
where
(65)n=1+2γ1+γ.


### 4.1. The Infiniteness of the Discrete Spectrum ofH

In this subsection we will derive the asymptotics (39) for the number of eigenvalues of H.We recall that in this subsection we closely follow Sobolev's method (see [8]) to derive the asymptotics for the number of eigenvalues of the operator H (see Theorem 10).LetT(δ;|z-Emin|) be the operator in ℋ(2) defined by
(66)T(δ;|z-Emin|)=(000T11(δ;|z-Emin|)),
where the T11(δ;|Emin-z|) is the integral operator in ℋ1 with the kernel
(67)T(δ,|Emin-z|;p,q)=-d0(((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4×(n(Wq,q)+2|Emin-z|)-1/4)×((1+γ)(Wp,p)+2γ(Wp,q)+(1+γ)(Wq,q)+2|Emin-z|)-1((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4),
where χ^δ(·) is the characteristic function of the region U^δ(0)={p∈𝕋3:|W1/2p|<δ} and
(68)d0=detW1/22π2(1+γ)3/2.Lemma 16.
Let the conditions of Theorem10 be fulfilled. The operator T(z)-T(δ;|Emin-z|) belongs to the Hilbert-Schmidt class and is continuous in z≤Emin.Proof.
Applying asymptotics (62) and (64), one can estimate the kernel of the operator T(z)-T(δ;|Emin-z|), z≤Emin, by the square-integrable function
(69)C(|Emin-z|1/2(p2+q2)-1(|p|2+|Emin-z|)1/4(|q|2+|Emin-z|)1/4|p|2+|q|2|p|1/2(p2+q2)|q|1/2+|Emin-z|1/2(p2+q2)-1(|p|2+|Emin-z|)1/4(|q|2+|Emin-z|)1/4+1).
Hence the operator T(z)-T(δ;|Emin-z|) belongs to the Hilbert-Schmidt class for all z≤Emin. In combination with the continuity of the kernel of the operator in z<Emin, this gives the continuity of T(z)-T(δ;|Emin-z|) in z≤Emin.LetT^0(δ;|Emin-z|) be the restriction of the integral operator T(δ;|Emin-z|) to the subspace L2(U^δ(0)). One verifies that the operator T^0(δ;|Emin-z|) is unitarily equivalent to the integral operator T1(r) acting in L2(Ur(0)), where r=|Emin-z|-1/2 and Ur(0)={p∈ℝ3:|p|<r}, with the kernel
(70)T1(r;p,q)=-d1(np2+2)-1/4(nq2+2)-1/4(1+γ)p2+2γ(p,q)+(1+γ)q2+2,
where
(71)d1=(1+γ)3/22π2.The equivalence is given by the unitary dilation(72)B:L2(U^δ(0))→L2(Ur(0)),(Brf)(p)=(rδ)-3/2f(δrW1/2p).Furthermore, we may replace(73)(np2+2)-1/4,(nq2+2)-1/4,(1+γ)p2+2γ(p,q)+(1+γ)q2+2
by
(74)(np2)-1/4(1-χ1(p)),(nq2)-1/4(1-χ1(q)),(1+γ)p2+2γ(p,q)+(1+γ)q2,
respectively, since the error will be a Hilbert-Schmidt operator continuous up to z=Emin.We have denoted byχ1(·) the characteristic function of the ball U1(0). By the replacement, we obtain the integral operator T2(r) in L2(Ur(0)∖U1(0)) with a kernel
(75)T2(r;p,q)=-d1n1/2|p|-1/2|q|-1/2(1+γ)p2+2γ(p,q)+(1+γ)q2.By the dilation(76)M:L2(Ur(0)∖U1(0))→L2((0,r),𝕊2),r=12|log|Emin-z||,
where 𝕊2 is the unit sphere in ℝ3, (Mf)(x,w)=e3x/2f(exw), x∈(0,r), w∈𝕊2, one sees that the operator T2(r) is unitarily equivalent to the integral operator Sr with the kernel Sγ(x-x′;<ξ,η>), ξ,η∈𝕊2, x,x′∈ℝ+, where
(77)Sγ(x;t)=-(2π)-2ucoshx+st,u=1+γ1+2γ,s=γ1+γ,t=〈ξ,η〉.For the completeness, we reproduce the following lemma, which has been proved in [8].Lemma 17.
LetA(z)=A0(z)+A1(z), where A0(A1) is compact and continuous in z<0(z≤0). Assume that for some function f(·), f(z)→0, z→-0 the limit
(78)limz→-0f(z)n(λ,A0(z))=l(λ)
exists and is continuous in λ>0. Then the same limit exists for A(z) and
(79)limz→-0f(z)n(λ,A(z))=l(λ).The following theorem is important for the proof of the asymptotics (39).Theorem 18.
Let the conditions of the part (ii) of Theorem10 be fulfilled. The following equalities
(80)lim|Emin-z|→0n(1,T1(|Emin-z|-1/2))|log|Emin-z||=limr→∞12r-1n(1,Sr)=𝒰0(γ),𝒰0(γ)>0,
hold.Proof.
The coefficient𝒰(γ) in the r.h.s of the asymptotics (80) will be expressed by means of the self-adjoint integral operator S^γ(y), y∈ℝ, in L2(𝕊2), whose kernel depends on the scalar product t=〈ξ,η〉 of the arguments ξ,η∈𝕊2 and has the form
(81)S^γ(t;y)=-(2π)-1usinh[yarccost]1-s2t2sinh(πy).
Forν>0, define
(82)𝒰(ν;γ)=(4π)-1∫-∞∞n(ν,S^γ(y))dy,ν>0.
The function𝒰(ν;γ) is very important for the proof of the existence of the Efimov effect. Denote 𝒰0(γ)=𝒰(1;γ).
Similarly to [8], we can derive that
(83)𝒰0(γ)=14π∫-∞∞‍n(1,S^γ(y))dy=14π∑l=0∞(2l+1)∫-∞∞‍n(1,S^γ(l)(y))dy,
where S^γ(l)(y) is the multiplication operator by number
(84)S^γ(l)(y)=2π∫-11‍Pl(t)S^γ(t;y)dt
in 𝔊l the subspace of the harmonics of degree l and Pl(t)  are Legendre polynomials. It follows from (83) and Lemma 19 that
(85)𝒰0(γ)≥14π∫-∞+∞n(1,S^γ(1)(y))dy=34πmes{x:S^γ(1)(x)>1}.
Lemma  19.   The following assertions are true.(a)
S^γ(0)(y)<0.(b)
supyS^γ(1)(y)>1,γ>γ*, andsupyS^γ(1)(y)<1,γ<γ*.(c)
S^γ(2)(y)<0,γ>0,
and for anyγ,γ<γ*there exists a positive numberε=εγsuch thatsupyS^γ(1)(y)<1-εγ,l≥3.
Proof. For the completeness of the proof of this lemma we reproduce in Appendix A.
The positivity of𝒰0(γ) follows from the fact that mes{x:S^γ(1)(x)>1}>0, if γ>γ*, which is proved in Lemma 19.
We remark that for allγ>γ0 the number 𝒰0(γ) is finite and
(86)𝒰0(γ)≤3πlog(π((1+γ)/1+2γ))π-2arcsin(γ/(1+γ))×[log(16(1+γ)/π1/233/41+2γ)log((1+γ)/γ+1+2γ/γ)+1]2.
The difference of the operators Sr and T1(r) is compact (up to unitarily equivalence) and hence, taking into account that r=|1/2|log|Emin-z| and Lemma 17, we obtain the limit (80).Proof of Part (ii) of Theorem10.
Theorem18 together with Lemmas 13, 16, and 17 complete the proof of Theorem 10.


### 4.2. The Finiteness of the Discrete Spectrum ofH

Lemma 20.
Letγ<γ* and the hypothesis of part (i) of Theorem 10 be fulfilled. Then there exists a number ϵ=ϵγ depending on γ, such that
(87)supσ<?brm?>ess<?erm?>(T(Emin))<1-ϵγ.Proof.
See AppendixB.Remark 21.
In the proof of Lemma20, we can see that the main part of the operator T(Emin) is unitarily equivalent to the direct sum of multiplication operators in L2(ℝ) by the functions S^γ(0)(·), l=0,1,…, in Lemma 19. So T(Emin) is not compact operator.By virtue of Lemma20 the operator T(Emin) cannot have many eigenvalues larger than 1, and hence
(88)n(1,T(Emin))<∞.This inequality and Lemma15 complete the proof of part (i) of Theorem 10.


## 4.1. The Infiniteness of the Discrete Spectrum ofH

In this subsection we will derive the asymptotics (39) for the number of eigenvalues of H.We recall that in this subsection we closely follow Sobolev's method (see [8]) to derive the asymptotics for the number of eigenvalues of the operator H (see Theorem 10).LetT(δ;|z-Emin|) be the operator in ℋ(2) defined by
(66)T(δ;|z-Emin|)=(000T11(δ;|z-Emin|)),
where the T11(δ;|Emin-z|) is the integral operator in ℋ1 with the kernel
(67)T(δ,|Emin-z|;p,q)=-d0(((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4×(n(Wq,q)+2|Emin-z|)-1/4)×((1+γ)(Wp,p)+2γ(Wp,q)+(1+γ)(Wq,q)+2|Emin-z|)-1((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4((n(Wq,q)+2|Emin-z|)-1/4χ^δ(p)χ^δ(q)(n(Wp,p)+2|Emin-z|)-1/4),
where χ^δ(·) is the characteristic function of the region U^δ(0)={p∈𝕋3:|W1/2p|<δ} and
(68)d0=detW1/22π2(1+γ)3/2.Lemma 16.
Let the conditions of Theorem10 be fulfilled. The operator T(z)-T(δ;|Emin-z|) belongs to the Hilbert-Schmidt class and is continuous in z≤Emin.Proof.
Applying asymptotics (62) and (64), one can estimate the kernel of the operator T(z)-T(δ;|Emin-z|), z≤Emin, by the square-integrable function
(69)C(|Emin-z|1/2(p2+q2)-1(|p|2+|Emin-z|)1/4(|q|2+|Emin-z|)1/4|p|2+|q|2|p|1/2(p2+q2)|q|1/2+|Emin-z|1/2(p2+q2)-1(|p|2+|Emin-z|)1/4(|q|2+|Emin-z|)1/4+1).
Hence the operator T(z)-T(δ;|Emin-z|) belongs to the Hilbert-Schmidt class for all z≤Emin. In combination with the continuity of the kernel of the operator in z<Emin, this gives the continuity of T(z)-T(δ;|Emin-z|) in z≤Emin.LetT^0(δ;|Emin-z|) be the restriction of the integral operator T(δ;|Emin-z|) to the subspace L2(U^δ(0)). One verifies that the operator T^0(δ;|Emin-z|) is unitarily equivalent to the integral operator T1(r) acting in L2(Ur(0)), where r=|Emin-z|-1/2 and Ur(0)={p∈ℝ3:|p|<r}, with the kernel
(70)T1(r;p,q)=-d1(np2+2)-1/4(nq2+2)-1/4(1+γ)p2+2γ(p,q)+(1+γ)q2+2,
where
(71)d1=(1+γ)3/22π2.The equivalence is given by the unitary dilation(72)B:L2(U^δ(0))→L2(Ur(0)),(Brf)(p)=(rδ)-3/2f(δrW1/2p).Furthermore, we may replace(73)(np2+2)-1/4,(nq2+2)-1/4,(1+γ)p2+2γ(p,q)+(1+γ)q2+2
by
(74)(np2)-1/4(1-χ1(p)),(nq2)-1/4(1-χ1(q)),(1+γ)p2+2γ(p,q)+(1+γ)q2,
respectively, since the error will be a Hilbert-Schmidt operator continuous up to z=Emin.We have denoted byχ1(·) the characteristic function of the ball U1(0). By the replacement, we obtain the integral operator T2(r) in L2(Ur(0)∖U1(0)) with a kernel
(75)T2(r;p,q)=-d1n1/2|p|-1/2|q|-1/2(1+γ)p2+2γ(p,q)+(1+γ)q2.By the dilation(76)M:L2(Ur(0)∖U1(0))→L2((0,r),𝕊2),r=12|log|Emin-z||,
where 𝕊2 is the unit sphere in ℝ3, (Mf)(x,w)=e3x/2f(exw), x∈(0,r), w∈𝕊2, one sees that the operator T2(r) is unitarily equivalent to the integral operator Sr with the kernel Sγ(x-x′;<ξ,η>), ξ,η∈𝕊2, x,x′∈ℝ+, where
(77)Sγ(x;t)=-(2π)-2ucoshx+st,u=1+γ1+2γ,s=γ1+γ,t=〈ξ,η〉.For the completeness, we reproduce the following lemma, which has been proved in [8].Lemma 17.
LetA(z)=A0(z)+A1(z), where A0(A1) is compact and continuous in z<0(z≤0). Assume that for some function f(·), f(z)→0, z→-0 the limit
(78)limz→-0f(z)n(λ,A0(z))=l(λ)
exists and is continuous in λ>0. Then the same limit exists for A(z) and
(79)limz→-0f(z)n(λ,A(z))=l(λ).The following theorem is important for the proof of the asymptotics (39).Theorem 18.
Let the conditions of the part (ii) of Theorem10 be fulfilled. The following equalities
(80)lim|Emin-z|→0n(1,T1(|Emin-z|-1/2))|log|Emin-z||=limr→∞12r-1n(1,Sr)=𝒰0(γ),𝒰0(γ)>0,
hold.Proof.
The coefficient𝒰(γ) in the r.h.s of the asymptotics (80) will be expressed by means of the self-adjoint integral operator S^γ(y), y∈ℝ, in L2(𝕊2), whose kernel depends on the scalar product t=〈ξ,η〉 of the arguments ξ,η∈𝕊2 and has the form
(81)S^γ(t;y)=-(2π)-1usinh[yarccost]1-s2t2sinh(πy).
Forν>0, define
(82)𝒰(ν;γ)=(4π)-1∫-∞∞n(ν,S^γ(y))dy,ν>0.
The function𝒰(ν;γ) is very important for the proof of the existence of the Efimov effect. Denote 𝒰0(γ)=𝒰(1;γ).
Similarly to [8], we can derive that
(83)𝒰0(γ)=14π∫-∞∞‍n(1,S^γ(y))dy=14π∑l=0∞(2l+1)∫-∞∞‍n(1,S^γ(l)(y))dy,
where S^γ(l)(y) is the multiplication operator by number
(84)S^γ(l)(y)=2π∫-11‍Pl(t)S^γ(t;y)dt
in 𝔊l the subspace of the harmonics of degree l and Pl(t)  are Legendre polynomials. It follows from (83) and Lemma 19 that
(85)𝒰0(γ)≥14π∫-∞+∞n(1,S^γ(1)(y))dy=34πmes{x:S^γ(1)(x)>1}.
Lemma  19.   The following assertions are true.(a)
S^γ(0)(y)<0.(b)
supyS^γ(1)(y)>1,γ>γ*, andsupyS^γ(1)(y)<1,γ<γ*.(c)
S^γ(2)(y)<0,γ>0,
and for anyγ,γ<γ*there exists a positive numberε=εγsuch thatsupyS^γ(1)(y)<1-εγ,l≥3.
Proof. For the completeness of the proof of this lemma we reproduce in Appendix A.
The positivity of𝒰0(γ) follows from the fact that mes{x:S^γ(1)(x)>1}>0, if γ>γ*, which is proved in Lemma 19.
We remark that for allγ>γ0 the number 𝒰0(γ) is finite and
(86)𝒰0(γ)≤3πlog(π((1+γ)/1+2γ))π-2arcsin(γ/(1+γ))×[log(16(1+γ)/π1/233/41+2γ)log((1+γ)/γ+1+2γ/γ)+1]2.
The difference of the operators Sr and T1(r) is compact (up to unitarily equivalence) and hence, taking into account that r=|1/2|log|Emin-z| and Lemma 17, we obtain the limit (80).Proof of Part (ii) of Theorem10.
Theorem18 together with Lemmas 13, 16, and 17 complete the proof of Theorem 10.


## 4.2. The Finiteness of the Discrete Spectrum ofH

Lemma 20.
Letγ<γ* and the hypothesis of part (i) of Theorem 10 be fulfilled. Then there exists a number ϵ=ϵγ depending on γ, such that
(87)supσ<?brm?>ess<?erm?>(T(Emin))<1-ϵγ.Proof.
See AppendixB.Remark 21.
In the proof of Lemma20, we can see that the main part of the operator T(Emin) is unitarily equivalent to the direct sum of multiplication operators in L2(ℝ) by the functions S^γ(0)(·), l=0,1,…, in Lemma 19. So T(Emin) is not compact operator.By virtue of Lemma20 the operator T(Emin) cannot have many eigenvalues larger than 1, and hence
(88)n(1,T(Emin))<∞.This inequality and Lemma15 complete the proof of part (i) of Theorem 10.



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