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

Abstract

We consider a model operator 𝐻 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 space ℱ_𝑎(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<𝐸_min=inf𝜎_ess(𝐻)∶lim_{𝑧→𝐸min}(𝑁(𝑧)/|log|𝐸_min−𝑧||)=𝒰₀(𝛾)(𝒰₀(𝛾)>0), for all 𝛾>𝛾^∗.

1. Introduction

In the spectral theory of the continuous and lattice three-particle Schrödinger operators in ℝ³, there is the remarkable phenomenon known as Efimov 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 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 [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 space ℱ_𝑠(𝐿²(𝕋³)) over 𝐿²(𝕋³).

In the present paper, we consider a model operator 𝐻, acting in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space ℱ_𝑎(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 operator 𝐻 with emphasis on the asymptotics for the number of infinitely many eigenvalues (Efimov's effect case).

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 (this parameter would be the ratio of the mass). Under some smoothness assumptions, we obtain the following results.(i)We find a critical value 𝛾^∗ 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 case 𝛾>𝛾^∗, we also establish the following asymptotics for the number 𝑁(𝑧) of eigenvalues below 𝑧<𝐸_min: lim_{𝑧→𝐸−min}𝑁(𝑧)log𝐸_min−𝑧=𝒰₀𝛾𝒰₀𝛾>0,∀𝛾>𝛾^∗. (1)

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 operator 𝐻 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. Section 1 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. Let 𝕋³=(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⊂ℝ^𝑛. Denote by 𝐿²_as((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 ℋ₀=ℂ,ℋ₁=𝐿²𝕋³,ℋ₂=𝐿²_as𝕋³². (2)

The Hilbert space ℋ⁽³⁾=ℋ₀⊕ℋ₁⊕ℋ₂ is called the direct sum of zero-, one-, and two-particle subspaces of a fermionic Fock space ℱ_𝑎(𝐿²(𝕋³)) over 𝐿²(𝕋³).

Let 𝐻_𝑖𝑗 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 that 𝐻_𝑖𝑗≡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 𝐻𝐻=⁰⁰𝐻₀₁0𝐻₁₀𝐻₁₁𝐻₁₂0𝐻₂₁𝐻₂₂, (3) where the operators 𝐻_𝑖𝑗∶ℋ_𝑗→ℋ_𝑖,  𝑖,𝑗=0,1,2 are defined by the forms 𝐻₀₀𝑓₀₀=𝑢₀𝑓₀,𝐻₀₁𝑓₁₀=_𝕋3𝑏𝑞^′𝑓₁𝑞^′𝑑𝑞^′,𝐻₁₀𝑓₀₁=𝑏𝑝𝑓₀,𝐻₁₁𝑓₁₁=𝑢𝑝𝑓₁𝑝,𝐻₁₂𝑓₂₁=_𝕋3𝑏𝑞^′𝑓₂𝑝,𝑞^′𝑑𝑞^′,𝐻₂₁𝑓₁₂=12𝑏𝑞𝑓₁𝑝−𝑏𝑝𝑓₁𝑞,𝐻₂₂𝑓₂₂=𝐸𝑝,𝑞𝑓₂𝑝,𝑞. (4)

Here 𝑓_𝑖∈ℋ_𝑖, 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 operator 𝐻 is bounded and self-adjoint in ℋ⁽³⁾.

Throughout this paper we assume the following additional technical assumptions.

Hypothesis 1. (a) The real-analytic function 𝐸(⋅,⋅) 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 𝜕²𝐸(0,0)𝜕𝑝_𝑖𝜕𝑝_𝑗³_𝑖,𝑗=1=𝜕1+𝛾𝑊,₂𝐸(0,0)𝜕𝑝_𝑖𝜕𝑞_𝑗³_𝑖,𝑗=1=𝛾𝑊. (5)(6) (b) The functions 𝑢(⋅) and 𝑏(⋅) on 𝕋³ are even and 𝑢(⋅) has a unique minimum at the origin.

By Hypothesis 1, for any 𝑝∈𝕋³ the integral _𝕋3𝑏²(𝑡)𝑑𝑡𝐸𝑝,𝑡−𝐸_min (7) is finite and hence it defines a continuous even function on 𝕋³, which will be denoted by Λ(𝑝).

Since the function 𝐸(𝑝,𝑞) has a unique nondegenerate minimum at the point (0,0)∈(𝕋³)² and 𝐸(0,0)=𝐸_min the function Λ(𝑝) is positive.

Hypothesis 2. For any nonzero 𝑝∈𝕋³, the inequality Λ(𝑝)<Λ(0) holds.

Remark 1. For the functions 𝑢𝑝𝑝𝑝𝑝,𝐸𝑝𝑞=𝜀+𝑐,𝑏=𝜀𝑝,𝑞=𝛾𝜀𝑝+𝑞+𝜀+𝜀,𝑝,𝑞∈𝕋³, (8) 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 function 𝜀∶𝕋^𝑚→ℂ is called conditionally negative definite if 𝜀(𝑝)=𝜀(−𝑝) and ^𝑛_𝑖,𝑗=1𝜀𝑝_𝑖−𝑝_𝑗𝑧_𝑖𝑧_𝑗≤0 (9) for all 𝑝₁,…,𝑝_𝑛∈𝕋^𝑚 and all 𝐳=(𝑧₁,…,𝑧_𝑛)∈ℂ^𝑛 satisfying ∑^𝑛_𝑖=1𝑧_𝑖=0.

2.1. The Friedrichs Model

To formulate the main results of the paper we introduce a family of Friedrichs models ℎ(𝑝), 𝑝∈𝕋³, which act in ℋ⁽²⁾≡ℋ₀⊕ℋ₁ with the entries ℎ₀₀𝑝𝑓₀₀=𝑢𝑝𝑓₀,ℎ₀₁=1√2𝐻₀₁,ℎ₁₀=ℎ^∗₀₁,ℎ₁₁𝑝𝑓₁₁𝑞=𝔢_𝑝𝑞𝑓₁𝑞, (10) where 𝔢_𝑝(⋅)=𝐸(𝑝,⋅).

Let the operator ℎ₀(𝑝), 𝑝∈𝕋³, act in ℋ⁽²⁾ as ℎ₀𝑝𝑓₀𝑓₁𝑞=0𝔢_𝑝𝑞𝑓₁𝑞. (11)

The perturbation ℎ(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 𝜎_ess(ℎ(𝑝)) of ℎ(𝑝) fills the following interval on the real axis: 𝜎_essℎ𝑝=𝑚𝑝,𝑀𝑝, (12) where 𝑚𝑝=min_𝑞∈𝕋3𝔢_𝑝𝑞,𝑀𝑝=max_𝑞∈𝕋3𝔢_𝑝𝑞. (13)

Definition 2. Let 𝑢(0)≠𝐸_min. The operator ℎ(0) is said to have a threshold resonance if the number 1 is an eigenvalue of the operator 𝑞=𝑏𝑞G𝜓2𝑢(0)−𝐸_min𝕋³𝑏(𝑡)𝜓(𝑡)𝑑𝑡𝔢₀(𝑡)−𝐸_min,𝜓∈𝐶𝕋³, (14) 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 any 𝑝∈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 Δ𝑝1𝑝,𝑧=𝑢−𝑧−2_𝕋³𝑏²(𝑡)𝑑𝑡𝐸.𝑝,𝑡−𝑧 (15) Since Δ(0,⋅) is continuous in 𝑧≤𝐸_min, the following finite limit exists: Δ0,𝐸_min=lim_{𝑧→𝐸min}Δ(0,𝑧). (16)

Lemma 4. For any 𝑝∈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 ℎ𝑝𝑓=𝑧𝑓,𝑓∈ℋ⁽²⁾,𝑧∈ℂ⧵𝑚𝑝,𝑀𝑝, (17) is equivalent to the following equation: ℎ₀₀𝑝−𝑧𝑓₀+ℎ₀₁𝑓₁𝑓=0,₁=−ℎ₁₁𝑝−𝑧⁻¹ℎ₁₀𝑓₀. (18)
Putting 𝑓₁ 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 Δ𝑝,𝑧𝑓₀=0. (19)
Thus (17) has nontrivial solution if and only if Δ(𝑝,𝑧)=0.

Lemma 5. (i) The operator ℎ(0) has a threshold resonance if and only if Δ(0,𝐸_min)=0 and 𝑏(0)≠0.(ii)If 𝑢(0)≤𝐸_min, then the operator ℎ(0) has no a threshold resonance.(iii)Assume Δ(0,𝐸_min)=0.(a)If 𝑏(0)≠0, then the operator ℎ(0) has a threshold resonance and the vector 𝑓=(𝑓₀,𝑓₁), where 𝑓₀𝑓=const≠0,₁𝑞𝑏𝑞𝑓=−⁰√2𝔢₀𝑞−𝐸_min∈𝐿¹𝕋³⧵𝐿²𝕋³, (20) obeys the equation ℎ(0)𝑓=𝐸_min𝑓.(b)If 𝑏(0)=0, then the number 𝑧=𝐸_min 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 operator ℎ(0) has a threshold energy resonance. Then by Definition 2 the number 1 is an eigenvalue of the operator (14) as 𝑞=𝑏𝑞G𝜓2𝑢(0)−𝐸_min𝕋³𝑏(𝑡)𝜓(𝑡)𝑑𝑡𝔢₀(𝑡)−𝐸_min,𝜓∈𝐶𝕋³, (21) and the associated eigenfunction 𝜓 satisfies 𝜓(0)≠0.
This solution is equal to the function 𝑏(⋅) (up to a constant factor) and hence Δ(0,𝐸_min)=0.
“If Part.” Let the equality Δ(0,𝐸_min)=0 hold and let 𝑏(0)≠0. Then the inequality 𝑢(0)≠𝐸_min 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)Since 𝑢(0)≤𝐸_min and −ℎ₀₁(ℎ₁₁(0)−𝐸_min)⁻¹ℎ₁₀<0, we get Δ(0,𝐸_min)<0 which ends the proof together with (ii).(iii)Similarly to the proof of Lemma 4, one can check that Δ(0,𝐸_min)=ℎ₀₀(0)−𝐸_min−ℎ₀₁(ℎ₁₁(0)−𝐸_min)⁻¹ℎ₁₀ holds, and it is a determinant of the equation ℎ₀₀𝑝−𝐸_min𝑓₀+ℎ₀₁𝑓₁𝑓=0,₁=−ℎ₁₁(0)−𝐸_min⁻¹ℎ₁₀𝑓₀,thatis𝑓₁𝑞𝑏𝑞𝑓=−⁰√2𝔢₀𝑞−𝐸_min⁻¹, (22)which is equivalent to the equation ℎ(0)𝑓=𝐸_min𝑓,𝑓∈ℂ⊕𝐿¹𝕋³. (23)
Thus (23) has nontrivial solution if and only if Δ(0,𝐸_min)=0.
By Hypothesis 1 there exist 𝑐, 𝐶>0 and a bounded, partially continuous 𝑏(⋅) function such that 𝑐𝑞²≤𝔢₀𝑞−𝐸_min≤𝐶𝑞²,𝑏𝑞=𝑏(0)+𝑏𝑞𝑞². (24)(a)If 𝑏(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)If 𝑏(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 𝐸_min.

Let 𝑊 be the 3×3 matrix defined in Hypothesis 1 and 𝑈_𝛿(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 operator ℎ(0) have a threshold resonance. Then for any 𝑝∈𝑈_𝛿(0), 𝛿>0 sufficiently small and 𝑧≤𝐸_min, the following decomposition Δ=4√𝑝,𝑧2𝜋²𝑏²(0)1+𝛾^3/2det(𝑊)^1/2𝑚𝑝−𝑧+Δ⁽⁰²⁾𝑚𝑝−𝑧+Δ⁽²⁰⁾𝑝,𝑧 (25) 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≤𝐸_min, resp.).

Lemma 7. Let the operator ℎ(0) have a threshold resonance. Then there exist positive numbers 𝑐, 𝐶 and 𝛿 such that 𝑐𝑝≤Δ𝑝,𝐸_min≤𝐶𝑝forany𝑝∈𝑈_𝛿(0),Δ𝑝,𝐸_min≥𝑐forany𝑝∈𝕋³⧵𝑈_𝛿(0). (26)(27)

Proof. By the assumption (a) of Hypothesis 1, we have 𝐸𝑝,𝑞=𝐸_min+𝐹𝑝,𝑞+𝑂𝑝⁴+𝑞⁴as𝑝,𝑞→0, (28) where 𝐹=1𝑝,𝑞21+𝛾𝑊𝑝,𝑝+2𝛾𝑊𝑝,𝑞+1+𝛾𝑊𝑞,𝑞. (29)
Since 𝐸(⋅,⋅) has a nondegenerate minimum at the origin and the equality min_𝑞∈𝕋3𝐹=1𝑝,𝑞21+2𝛾1+𝛾𝑊𝑝,𝑝 (30) Holds, we have 𝑚𝑝=min_𝑞∈𝕋3𝐸𝑝,𝑞=𝐸_min+121+2𝛾𝑝1+𝛾𝑊𝑝,𝑝+𝑂⁴as𝑝→0. (31)
Consequently, these asymptotics and Lemma 6 yield (26) for some positive numbers 𝑐, 𝐶.
By Hypotheses 1 and 2, we have 𝑢(𝑝)>𝑢(0) and Λ(𝑝)>Λ(0), 0≠𝑝∈𝕋³, and hence Δ𝑝,𝐸_min=𝑢𝑝−𝐸_min−12Λ𝑝>Δ0,𝐸_min,0≠𝑝∈𝕋³. (32)
Then the inequality (27) follows from Δ(0,𝐸_min)=0 and the continuity of the function Δ(𝑝,𝐸_min).

2.2. The Essential Spectrum of 𝐻

The following theorem describes the essential spectrum of the operator 𝐻. 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 spectrum 𝜎_{<?brm?>ess<?erm?>}(𝐻) of the operator 𝐻, the equality 𝜎_{<?brm?>ess<?erm?>}(𝐻)=_𝑝∈𝕋3𝜎_𝑑ℎ𝑝∪𝐸_min,𝐸_max (33) holds, where 𝜎_𝑑(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 operator 𝐻.

Lemma 9. Let Hypothesis 1 be fulfilled and ℎ(0) has a threshold resonance. Then 𝜎_{<?brm?>ess<?erm?>}(𝐻)=𝐸_min,𝐸_max∪𝜎₂, (34) where 𝜎₂={𝑧∈[𝐸_max,+∞)∶Δ(𝑝,𝑧)=0forsome𝑝∈𝕋³}.

Proof. According to Lemma 4, the zero of Δ(𝑝,⋅) is the eigenvalues of ℎ(𝑝), so Υ=_𝑝∈𝕋3𝜎_𝑑ℎ𝑝⧵𝐸_min,𝐸_max=𝑧∈ℝ⧵𝐸_min,𝐸_max∶Δ𝑝,𝑧=0forsome𝑝∈𝕋³. (35)
By analyticity of Δ(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 function Δ(𝑝,⋅) is decreasing in (−∞,𝐸_min), respectively, with (𝐸_max,+∞), and hence there may exist its unique zero lying below, respectively, above the closed interval [𝐸_min,𝐸_max].
So, the set Υ may be described as the union of intervals 𝜎₁ and 𝜎₂ lying below and above of [𝐸_min,𝐸_max], and 𝜎₁=𝑧∈−∞,𝐸_min∶Δ𝑝,𝑧=0forsome𝑝∈𝕋³,𝜎₂=𝑧∈𝐸_max,+∞∶Δ𝑝,𝑧=0forsome𝑝∈𝕋³. (36)
By virtue of Hypothesis 2, we have Δ(𝑝,𝐸_min)≥Δ(0,𝐸_min) and using Lemma 5 we get Δ(0,𝐸_min)=0, and so Δ(𝑝,𝑧)>Δ(𝑝,𝐸_min)≥Δ(𝑝,𝐸_min)=0, that is Δ𝑝,𝑧>0∀𝑧<𝐸_min, (37) since Δ(𝑝,⋅) is decreasing in (−∞,𝐸_min). Then according to Lemma 4 we get 𝜎₁=∅.
Consequently, ∪_𝑝∈𝕋³𝜎_𝑑(ℎ(𝑝))⧵(𝐸_min,𝐸_max)=𝜎₂ and Theorem 8 ends the proof of the lemma.

2.3. Statement of the Main Results

Henceforth we assume that ℎ(0) has a threshold resonance.

According to Lemma 9 for the bottom of the essential spectrum the equality 𝐸_min=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≤𝐸_min.

Let 𝛾^∗ be a solution of the equation 21+𝛾²𝜋𝛾−21+2𝛾1+𝛾²𝜋𝛾²𝛾arcsin1+𝛾=1,𝛾>0. (38)

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 of 𝐻 on the parameters 𝛾>0.

Theorem 10. Let Hypothesis 1 be fulfilled and let ℎ(0) have a threshold resonance at 𝐸_min. Then we have the following. (i)For any 0<𝛾<𝛾^∗, the operator 𝐻 has a finite number of eigenvalues lying below the bottom 𝐸_min of the essential spectrum.(ii)For any 𝛾>𝛾^∗, the operator 𝐻 has infinitely many eigenvalues lying below 𝐸_min. The function 𝑁(𝑧) obeys the relation lim_{𝑧→𝐸−min}𝑁(𝑧)log𝐸_min−𝑧=𝒰₀𝛾𝒰₀𝛾>0. (39)

It follows from the positivity of the limit (39) that the discrete spectrum of the operator 𝐻 is infinite.

Remark 11. The constant 𝒰₀(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 ℤ^𝑑.

3. The Birman-Schwinger Principle

For a bounded self-adjoint operator 𝐵, we define 𝑛(𝜆,𝐵) by 𝑛(𝜆,𝐵)=sup{dim𝐹∶(𝐵𝑢,𝑢)>𝜆,𝑢∈𝐹,||𝑢||=1}. (40)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 𝑁(𝑧)=𝑛(−𝑧,−𝐻),−𝑧>−𝐸_min. (41)

In our analysis of the spectrum of 𝐻, the crucial role is played by the self-adjoint compact Faddeev-Newton-type integral operator 𝑇(𝑧), 𝑧<𝐸_min, in the space ℋ⁽²⁾ with the entries 𝑇₀₀(𝑧)𝑓₀₀=1−𝑢₀−𝑧𝑓₀,𝑇₀₁(𝑧)𝑓₁₀=−_𝕋3𝑏𝑞^′𝑓₁𝑞^′𝑑𝑞^′Δ𝑞^′,𝑇,𝑧₁₀(𝑧)=𝑇^∗₀₁(𝑧),𝑇₁₁(𝑧)𝑓₁₁𝑝𝑏𝑝=−2Δ𝑝,𝑧^𝕋3𝑏𝑞^′𝑓₁𝑞^′𝑑𝑞^′Δ𝑞^′,𝑧𝐸𝑝,𝑞^′.−𝑧 (42)

Now we describe 𝑇(𝑧) with the entries of 𝐻 in (3). We split 𝐻₂₁ into the sum of two operators 𝐻₂₁=12𝐻⁽¹⁾₂₁−𝐻⁽²⁾₂₁,𝐻⁽¹⁾₂₁𝑓₁₂=𝑏𝑞𝑓₁𝑝,𝐻⁽²⁾₂₁𝑓₁₂=𝑏𝑝𝑓₁𝑞, (43) where 𝑓₁∈ℋ₁.

According to the last representation and (3), we take 𝑇₀₀(𝑧)=(1+𝑧)𝐼₀−𝐻₀₀,𝑇₀₁(𝑧)=−𝐻₀₁𝐷^−1/2(𝑧),𝑇₁₀(𝑧)=−𝐷^−1/2(𝑧)𝐻₁₀,𝑇₁₁(𝑧)1=−2𝐷^−1/2(𝑧)𝐻₁₂𝑅^1/2₂₂(𝑧)𝐻⁽²⁾₂₁𝐷^−1/2(𝑧), (44) where 𝑅₂₂(𝑧)=(𝐻₂₂−𝑧𝐼₂)⁻¹ is the resolvent of 𝐻₂₂, 𝐷(𝑧)=𝐻₁₁−𝑧𝐼₁−12𝐷^−1/2(𝑧)𝐻₁₂𝑅^1/2₂₂(𝑧)𝐻⁽¹⁾₂₁𝐷^−1/2(𝑧). (45)

One can verify that 𝐷(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 operator 𝑇(𝑧) is compact and continuous in 𝑧<𝐸_min and 𝑁(𝑧)=𝑛(1,𝑇(𝑧)). (46)

Proof. Set 𝑀(𝑧)=ℛ^1/2(𝑧)ℛ⁻¹(𝑧)−(𝐻−𝑧𝐼)ℛ^1/2(𝑧), (47) where ℛ(𝑧)=diag{𝐼₀,𝐼₁,ℛ₂₂(𝑧)} and ℛ⁻¹(𝑧) is the inverse of ℛ(𝑧).
For entries of matrix operator 𝑀(𝑧), we have 𝑀₀₀(𝑧)=(1+𝑧)𝐼₀−𝐻₀₀,𝑀₀₁(𝑧)=−𝐻₀₁,𝑀₀₂(𝑧)𝑀=0,₁₀(𝑧)=−𝐻₁₀,𝑀₁₁(𝑧)=(1+𝑧)𝐼₁−𝐻₁₁,𝑀₁₂(𝑧)=−𝐻₁₂𝑅^1/2₂₂(𝑧),𝑀₂₀(𝑧)=0,𝑀₂₁(𝑧)=−𝑅^1/2₂₂(𝑧)𝐻₂₁,𝑀₂₂(𝑧)=0. (48)
The correctness of 𝑁(𝑧)=𝑛(1,𝑀(𝑧)) (49) 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∈ℋ_𝑀(𝑧)(1), where 𝑔=ℛ^−1/2(𝑧)𝑔.
For any 𝑧<𝜏_{<?brm?>ess<?erm?>}(𝐻), we define an operator 𝑉(𝑧) acting in the Hilbert space ℋ₀⊕ℋ₁ by the form 𝑉(𝑧)=𝑀₀₀(𝑧)𝑀₀₁(𝑧)𝑀₁₀(𝑧)𝑀₁₁(𝑧)+𝑀₁₂(𝑧)𝑀₂₁(𝑧), (50) and prove the equality 𝑁(𝑧)=𝑛(1,𝑉(𝑧)),𝑧<𝜏_{<?brm?>ess<?erm?>}(𝐻). (51)
Let 𝑓=(𝑓₀,𝑓₁)∈ℋ_𝑉(𝑧)(1), and then for 𝑔=(𝑓₀,𝑓₁,−𝑀₂₁(𝑧)𝑓₁)∈ℋ⁽³⁾, we get 𝑀(𝑧)𝑔,𝑔=𝑉(𝑧)𝑓,𝑓+𝑀₂₁(𝑧)𝑓₁,𝑔₂>𝑓,𝑓+𝑔₂,𝑔₂=𝑔,𝑔, (52) that is, 𝑔∈ℋ_𝑀(𝑧)(1).
Therefore, 𝑛(1,𝑉(𝑧))≤𝑛(1,𝑀(𝑧)),𝑧<𝜏_{<?brm?>ess<?erm?>}(𝐻). (53)
Using the equality 𝑉(𝑧)𝑓,𝑓=𝑀(𝑧)𝑔,𝑔+𝑀₁₂(𝑧)𝑀₂₁(𝑧)𝑔₁,𝑔₁−𝑀₂₁(𝑧)𝑔₁,𝑔₂−𝑀₁₂(𝑧)𝑔₂,𝑔₁, (54) where 𝑔∈ℋ⁽³⁾, 𝑓=(𝑔₁,𝑔₂)∈ℋ⁽²⁾, and taking 𝑔=(𝑔₀,𝑔₁,𝑔₂)∈ℋ_𝑀(𝑧)(1) realized analogously, we get 𝑉(𝑧)𝑓,𝑓>𝑓,𝑓+𝑔₂−𝑀₂₁(𝑧)𝑔₁²≥𝑓,𝑓,thatis,𝑓∈ℋ_𝑉(𝑧)(1) (55) and so 𝑛(1,𝑉(𝑧))≥𝑛(1,𝑀(𝑧)),𝑧<𝜏_{<?brm?>ess<?erm?>}(𝐻). (56)
Observe that 𝑉(𝑧)−𝐼=𝒟^1/2(𝑧)(𝑇(𝑧)−𝐼)𝒟^1/2(𝑧), (57) where 𝒟^1/2(𝑧) is acting in the Hilbert space ℋ₀⊕ℋ₁ by the equality 𝒟^1/2(𝑧)=diag𝐼₀,𝒟^1/2(𝑧). (58)
Employing (57), we show 𝑔∈ℋ_𝑉(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∈ℋ_𝑇(𝑧)(1).
Consequently, for any 𝑧<𝜏_{<?brm?>ess<?erm?>}(𝐻), we have 𝑛(1,𝑉(𝑧))=𝑛(1,𝑇(𝑧)),thatis𝑁(𝑧)=𝑛(1,𝑇(𝑧)). (59)

3.1. The Birman-Schwinger Principle at the Threshold

It should be noted that the operator 𝑇(𝑧) can be defined as a bounded operator even for the point 𝑧=𝐸_min by 𝑇₀₀𝐸_min𝑓₀₀=1−𝑢₀−𝐸_min𝑓₀,𝑇₀₁𝐸_min𝑓₁₀=−_𝕋3𝑏𝑞^′𝑓₁𝑞^′𝑑𝑞^′Δ𝑞^′,𝑇,𝑧₁₀𝐸_min=𝑇^∗₀₁𝐸_min,𝑇₁₁𝐸_min𝑓₁₁𝑝𝑏𝑝=−2Δ𝑝,𝐸_min^𝕋3𝑏𝑞^′𝑓₁𝑞^′𝑑𝑞^′Δ𝑞^′,𝐸_min𝐸𝑝,𝑞^′−𝐸_min. (60)

Remark 14. The operator 𝑇(𝑧) converges strongly (but not uniformly) as 𝑧→𝐸_min−0 to 𝑇(𝐸_min). 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 Lemma 13.

Lemma 15. For any 𝑧, 𝑧≤𝐸_min, the inequality 𝑁(𝑧)≤𝑛1,𝑇𝐸_min (61) occurs.

4. The Sketch of the Proof of the Main Results

By Hypothesis 1, we get 𝐸𝑝,𝑞=𝐸_min+12+𝑝1+𝛾𝑊𝑝,𝑝+2𝛾𝑊𝑝,𝑞1+𝛾𝑊𝑞,𝑞+𝑂⁴+𝑞⁴ (62) as 𝑝,𝑞→0 and (31) as 𝑚(𝑘)=𝐸_min+121+2𝛾()|𝑘|1+𝛾𝑊𝑘,𝑘+𝑂⁴as𝑘→0. (63)

Applying the asymptotics for 𝑚(𝑝) and using Lemma 6, we have Δ=𝑝,𝑧4𝜋²𝑏²(0)1+𝛾^3/2det(𝑊)^1/2𝑛𝑊𝑝,𝑝+2𝐸_min−𝑧^1/2+𝑂𝑝²+𝐸_min−𝑧as𝑝,𝐸_min−𝑧→0, (64) where 𝑛=1+2𝛾.1+𝛾 (65)

4.1. The Infiniteness of the Discrete Spectrum of 𝐻

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).

Let 𝑇(𝛿;|𝑧−𝐸_min|) be the operator in ℋ⁽²⁾ defined by 𝑇𝛿;𝑧−𝐸_min=000𝑇₁₁𝛿;𝑧−𝐸_min, (66) where the 𝑇₁₁(𝛿;|𝐸_min−𝑧|) is the integral operator in ℋ₁ with the kernel 𝑇𝛿,𝐸_min−𝑧;𝑝,𝑞=−𝑑₀𝜒_𝛿𝑝𝜒_𝛿𝑞𝑛𝑊𝑝,𝑝+2𝐸_min−𝑧^−1/4×𝑛𝑊𝑞,𝑞+2𝐸_min−𝑧^−1/4×+𝐸1+𝛾𝑊𝑝,𝑝+2𝛾𝑊𝑝,𝑞1+𝛾𝑊𝑞,𝑞+2_min−𝑧⁻¹, (67) where 𝜒_𝛿(⋅) is the characteristic function of the region 𝑈_𝛿(0)={𝑝∈𝕋³∶|𝑊^1/2𝑝|<𝛿} and 𝑑₀=det𝑊^1/22𝜋²1+𝛾^3/2. (68)

Lemma 16. Let the conditions of Theorem 10 be fulfilled. The operator 𝑇(𝑧)−𝑇(𝛿;|𝐸_min−𝑧|) belongs to the Hilbert-Schmidt class and is continuous in 𝑧≤𝐸_min.

Proof. Applying asymptotics (62) and (64), one can estimate the kernel of the operator 𝑇(𝑧)−𝑇(𝛿;|𝐸_min−𝑧|), 𝑧≤𝐸_min, by the square-integrable function 𝐶𝑝²+𝑞²𝑝^1/2𝑝²+𝑞²𝑞^1/2+𝐸_min−𝑧^1/2𝑝²+𝑞²⁻¹𝑝²+𝐸_min−𝑧^1/4𝑞²+𝐸_min−𝑧^1/4+1. (69) Hence the operator 𝑇(𝑧)−𝑇(𝛿;|𝐸_min−𝑧|) belongs to the Hilbert-Schmidt class for all 𝑧≤𝐸_min. In combination with the continuity of the kernel of the operator in 𝑧<𝐸_min, this gives the continuity of 𝑇(𝑧)−𝑇(𝛿;|𝐸_min−𝑧|) in 𝑧≤𝐸_min.

Let 𝑇₀(𝛿;|𝐸_min−𝑧|) be the restriction of the integral operator 𝑇(𝛿;|𝐸_min−𝑧|) to the subspace 𝐿²(𝑈_𝛿(0)). One verifies that the operator 𝑇₀(𝛿;|𝐸_min−𝑧|) is unitarily equivalent to the integral operator 𝑇₁(𝑟) acting in 𝐿²(𝑈_𝑟(0)), where 𝑟=|𝐸_min−𝑧|^−1/2 and 𝑈_𝑟(0)={𝑝∈ℝ³∶|𝑝|<𝑟}, with the kernel 𝑇₁𝑟;𝑝,𝑞=−𝑑₁𝑛𝑝²+2^−1/4𝑛𝑞²+2^−1/41+𝛾𝑝²+2𝛾𝑝,𝑞+1+𝛾𝑞²,+2 (70) where 𝑑₁=1+𝛾^3/22𝜋². (71)

The equivalence is given by the unitary dilation 𝐵∶𝐿²𝑈_𝛿(0)→𝐿²𝑈_𝑟(0,𝐵_𝑟)𝑓𝑝=𝑟𝛿^−3/2𝑓𝛿𝑟𝑊^1/2𝑝. (72)

Furthermore, we may replace 𝑛𝑝²+2^−1/4,𝑛𝑞²+2^−1/4,1+𝛾𝑝²+2𝛾𝑝,𝑞+1+𝛾𝑞²+2 (73) by 𝑛𝑝^2−1/41−𝜒₁𝑝,𝑛𝑞^2−1/41−𝜒₁𝑞,1+𝛾𝑝²+2𝛾𝑝,𝑞+1+𝛾𝑞², (74) respectively, since the error will be a Hilbert-Schmidt operator continuous up to 𝑧=𝐸_min.

We have denoted by 𝜒₁(⋅) the characteristic function of the ball 𝑈₁(0). By the replacement, we obtain the integral operator 𝑇₂(𝑟) in 𝐿²(𝑈_𝑟(0)⧵𝑈₁(0)) with a kernel 𝑇₂𝑑𝑟;𝑝,𝑞=−¹𝑛^1/2𝑝^−1/2𝑞^−1/21+𝛾𝑝²+2𝛾𝑝,𝑞+1+𝛾𝑞². (75)

By the dilation 𝑀∶𝐿²𝑈_𝑟(0)⧵𝑈₁(0)→𝐿²(0,𝐫),𝕊²,1𝐫=2log𝐸_min−𝑧, (76) where 𝕊² is the unit sphere in ℝ³, (𝑀𝑓)(𝑥,𝑤)=𝑒^3𝑥/2𝑓(𝑒^𝑥𝑤), 𝑥∈(0,𝐫), 𝑤∈𝕊², one sees that the operator 𝑇₂(𝑟) is unitarily equivalent to the integral operator 𝐒_𝐫 with the kernel 𝑆_𝛾(𝑥−𝑥^′;<𝜉,𝜂>), 𝜉,𝜂∈𝕊², 𝑥,𝑥^′∈ℝ⁺, where 𝑆_𝛾(𝑥;𝑡)=−(2𝜋)⁻²𝑢,cosh𝑥+𝑠𝑡𝑢=1+𝛾𝛾1+2𝛾,𝑠=.1+𝛾,𝑡=𝜉,𝜂 (77)

For the completeness, we reproduce the following lemma, which has been proved in [8].

Lemma 17. Let 𝐴(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 lim_𝑧→−0𝑓(𝑧)𝑛𝜆,𝐴₀(𝑧)=𝑙(𝜆) (78) exists and is continuous in 𝜆>0. Then the same limit exists for 𝐴(𝑧) and lim_z→−0𝑓(𝑧)𝑛(𝜆,𝐴(𝑧))=l(𝜆). (79)

The following theorem is important for the proof of the asymptotics (39).

Theorem 18. Let the conditions of the part (ii) of Theorem 10 be fulfilled. The following equalities lim_{𝐸min−𝑧→0}𝑛1,𝑇₁𝐸_min−𝑧^−1/2log𝐸_min−𝑧=lim_𝐫→∞12𝐫⁻¹𝑛1,𝐒_𝐫=𝒰₀𝛾,𝒰₀𝛾>0, (80) hold.

Proof. The coefficient 𝒰(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 𝑆_𝛾𝑡;𝑦=−(2𝜋)⁻¹𝑢sinh𝑦arccos𝑡√1−𝑠²𝑡².sinh𝜋𝑦 (81)
For 𝜈>0, define 𝒰𝜈;𝛾=(4𝜋)^−1∞_−∞𝑛𝜈,𝐒_𝛾𝑦𝑑𝑦,𝜈>0. (82)
The function 𝒰(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 𝒰₀𝛾=14𝜋^∞_−∞𝑛1,𝐒_𝛾𝑦=1𝑑𝑦4𝜋^∞_𝑙=0(2𝑙+1)^∞_−∞𝑛1,𝐒_𝛾^(𝑙)𝑦𝑑𝑦, (83) where 𝐒_𝛾^(𝑙)(𝑦) is the multiplication operator by number 𝑆_𝛾^(𝑙)𝑦=2𝜋¹₋₁𝑃_𝑙(𝑡)𝑆_𝛾𝑡;𝑦𝑑𝑡 (84) in 𝔊_𝑙 the subspace of the harmonics of degree 𝑙 and 𝑃_𝑙(𝑡) are Legendre polynomials. It follows from (83) and Lemma 19 that 𝒰₀𝛾≥14𝜋^+∞_−∞𝑛1,𝐒_𝛾⁽¹⁾𝑦3𝑑𝑦=𝑆4𝜋mes𝑥∶_𝛾⁽¹⁾(𝑥)>1. (85) Lemma  19.  The following assertions are true. (a)𝑆_𝛾⁽⁰⁾(𝑦)<0. (b)sup_𝑦𝑆_𝛾⁽¹⁾(𝑦)>1, 𝛾>𝛾^∗, and sup_𝑦𝑆_𝛾⁽¹⁾(𝑦)<1, 𝛾<𝛾^∗.(c)𝑆_𝛾⁽²⁾(𝑦)<0, 𝛾>0, and for any 𝛾, 𝛾<𝛾^∗ there exists a positive number 𝜀=𝜀_𝛾 such that sup_𝑦𝑆_𝛾⁽¹⁾(𝑦)<1−𝜀_𝛾, 𝑙≥3.
Proof. For the completeness of the proof of this lemma we reproduce in Appendix A.
The positivity of 𝒰₀(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 all 𝛾>𝛾₀ the number 𝒰₀(𝛾) is finite and 𝒰₀𝛾≤3𝜋log𝜋1+𝛾/1+2𝛾×𝜋−2arcsin𝛾/1+𝛾log161+𝛾/𝜋^1/23^3/41+2𝛾log1+𝛾/𝛾+1+2𝛾/𝛾+1². (86) The difference of the operators 𝐒_𝐫 and 𝑇₁(𝑟) is compact (up to unitarily equivalence) and hence, taking into account that 𝐫=|1/2|log|𝐸_min−𝑧| and Lemma 17, we obtain the limit (80).

Proof of Part (ii) of Theorem 10. Theorem 18 together with Lemmas 13, 16, and 17 complete the proof of Theorem 10.

4.2. The Finiteness of the Discrete Spectrum of 𝐻

Lemma 20. Let 𝛾<𝛾^∗ and the hypothesis of part (i) of Theorem 10 be fulfilled. Then there exists a number 𝜖=𝜖_𝛾 depending on 𝛾, such that sup𝜎_{<?brm?>ess<?erm?>}𝑇𝐸_min<1−𝜖_𝛾. (87)

Proof. See Appendix B.

Remark 21. In the proof of Lemma 20, we can see that the main part of the operator 𝑇(𝐸_min) 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(𝐸_min) is not compact operator.

By virtue of Lemma 20 the operator 𝑇(𝐸_min) cannot have many eigenvalues larger than 1, and hence 𝑛1,𝑇𝐸_min<∞. (88)

This inequality and Lemma 15 complete the proof of part (i) of Theorem 10.

Appendices

A. Proof of Lemma  19

Case (𝑙=0). By (83) we first calculate 𝑆_𝛾⁽⁰⁾(𝑦) as 𝑆_𝛾⁽⁰⁾𝑦𝑢=−sinh𝜋𝛾¹₋₁sinh𝑦(arccos(𝑠𝑡))√1−𝑠²𝑡²𝑑𝑡. (A.1) In [8] it is proven that for any 𝑦∈ℝ, 𝑢sinh𝑦(arcsin𝑠)=𝑢𝑠𝑦cosh𝜋𝑦sinh𝜋𝛾¹₋₁sinh𝑦(arccos(𝑠𝑡))√1−𝑠²𝑡²𝑑𝑡. (A.2) Since this number is positive by (A.1) we have 𝑆_𝛾⁽⁰⁾(𝑦)<0 for all 𝑦∈ℝ.

Case (𝑙=1). For any 𝑦∈ℝ¹, the 𝑆_𝛾⁽¹⁾(𝑦) can be written in the form 𝑆_𝛾⁽¹⁾𝑦=𝑢sinh𝜋𝑦/2¹₀𝑡sinh𝑦(arcsin(𝑠𝑡))√1−𝑠²𝑡²𝑑𝑡. (A.3)

Since the integrand in (A.3) is positive, we obtain that the function 𝑆_𝛾⁽¹⁾(𝑦) is also positive and even as 𝑦∈ℝ¹. The function sinh𝛽𝑦/sinh𝛼𝑦 is strictly decreased on [0,∞) for 𝛼>𝛽 and hence the function 𝑆_𝛾⁽¹⁾(𝑦) strictly decreases on [0,∞).

Let 𝑏₁(𝛾)=sup_𝑦𝑆_𝛾⁽¹⁾(𝑦). Then we have 𝑏₁𝛾=21+𝛾²𝜋𝛾−21+2𝛾1+𝛾²𝜋𝛾²𝛾arcsin.1+𝛾 (A.4)

Since 𝑆_𝛾⁽¹⁾(𝑦)>0 for all 𝑦∈ℝ¹ and lim_𝑦→∞𝑆_𝛾⁽¹⁾(𝑦)=0, it follows from (A.3) and (A.4) that the range of the function 𝑆_𝛾⁽¹⁾(𝑦) coincides with the set (0,𝑏₁(𝛾)].

The function 𝑏₁(𝛾) defined on (0,∞) is continuous and strictly increases.

Moreover, lim_𝛾→0𝑏₁𝛾=0,lim_𝛾→∞𝑏₁𝛾=∞. (A.5) Thus the equation 𝑏₁(𝛾)=1 has a unique simple solution 𝛾₀>0 and 𝑏₁(𝛾)<1 (𝑏₁(𝛾)>1, resp.) for 𝛾<𝛾^∗ (𝛾>𝛾^∗, resp.).

Case (𝑙=2). The function 𝑆_𝛾⁽²⁾(𝑦) is calculated by 𝑆_𝛾⁽²⁾𝑦𝑢=−2¹₀3𝑡²−1√1−𝑠²𝑡²cosh𝑦(arcsin𝑠𝑡)cosh𝜋𝑦/2𝑑𝑡. (A.6) It is easy to see that the inequalities cosh𝑦(arcsin𝑠𝑡)√1−𝑠²𝑡²<cosh𝑦arcsin1/√3𝑠1−(1/3)𝑠²1,𝑡∈0,√3,cosh𝑦(arcsin𝑠𝑡)√1−𝑠²𝑡²>cosh𝑦arcsin1/√3𝑠1−(1/3)𝑠²1,𝑡∈√3,1, (A.7) hold.

Then using the inequalities (A.7), we obtain the following inequality: 𝑆_𝛾⁽²⁾𝑦𝑢<−2cosh𝑦arcsin1/√3𝑠1−(1/3)𝑠²cosh𝜋𝑦/2¹⁰3𝑡²−1𝑑𝑡=0. (A.8) From here it follows that for all 𝛾>0sup𝜎𝑆_𝛾⁽²⁾𝑦𝑑𝑦≤0. (A.9)

Case (𝑙≥3). By Lemma 3.2 of [8], we get that 𝑆_𝛾^(𝑙)𝑦=<𝐹𝑙;𝛾16𝜋^1/2[3(2𝑙+1)]^3/41+𝛾𝛾1+2𝛾1+𝛾+1+2𝛾^𝑙 (A.10) for all 𝑙≥0 and 𝛾>0.

Since the function 𝐹(𝑙;⋅) is increasing and the 𝐹(⋅;𝛾) is decreasing, we have sup_𝑦𝑆_𝛾^(𝑙)𝑦<𝐹𝑙;𝛾^∗<𝐹3;𝛾^∗<𝐹(3;16)<1 (A.11)

for any 𝛾≤𝛾^∗ and 𝑙≥3.

Thus for any 𝛾≥0 and 𝑦∈ℝ¹ we have 𝑆_𝛾^(𝑙)(𝑦)≤0, 𝑙=0,2 and for any 𝛾<𝛾^∗ (for any 𝛾≥0 and 𝑙≥3, resp.) we have |𝑆_𝛾⁽¹⁾(𝑦)|<1 (resp. |𝑆_𝛾^(𝑙)(𝑦)|<1).

These facts complete the proof of the lemma.

B. Proof of Lemma 20

Since the operator 𝑇(𝐸_min)−𝑇(𝛿;0) is Hilbert-Schmidt (see Lemma 16), Weyl's Theorem implies that 𝜎_{<?brm?>ess<?erm?>}(𝑇(𝐸_min))=𝜎_{<?brm?>ess<?erm?>}(𝑇(𝛿;0)).

The space 𝐿²(𝑈_𝛿(0)) of all functions 𝑤(𝑝) having support in 𝑈_𝛿(0)={𝑝∈𝕋³∶|𝑊^1/2𝑝|<𝛿} is an invariant subspace of the operator 𝑇_𝛾(𝛿;0).

Let 𝑇_𝛾(𝛿;0) be the restriction of the operator 𝑇_𝛾(𝛿;0) to the invariant subspace 𝐿²(𝑈_𝛿(0)).

The operator 𝑇_𝛾(𝛿;0) is unitary equivalent to the integral operator 𝐒_𝛾 acting in 𝐿²(0,+∞)⊗𝐿²(𝕊²), with the kernel 𝑆_𝛾(𝑥−𝑥^′;<𝜉,𝜂>), 𝜉, 𝜂∈𝕊², where 𝑆_𝛾(𝑥;𝑡)=−(2𝜋)⁻²𝑢cosh𝑥+𝑠𝑡,𝑢=1+𝛾,𝛾1+2𝛾𝑠=.1+𝛾 (B.1)

The equivalence is given by the unitary operator 𝑀∶𝐿²(𝑈_𝛿(0))→𝐿²((0,∞)⊗𝐿²(𝕊²)), where 𝑀𝑓(𝑥,𝑤)=𝛿^3/2𝑒^−3𝑥/2𝑓𝛿𝑒^−𝑥𝑊12𝑤,𝑤∈𝕊². (B.2)

Therefore, the essential spectrum of 𝑇_𝛾⁽⁰⁾(𝛿) and 𝐒_𝛾 coincide.

Similarly to [8], we have 𝐒_𝛾=^∞_𝑙=0⊕𝐒_𝛾^(𝑙)⊗𝒫_𝑙, (B.3) where 𝒫_𝑙∶𝐿²((0,∞)⊗𝐿²(𝕊²))→𝔊_𝑙 is the orthogonal projector onto 𝔊_𝑙, the operator 𝐒_𝛾^(𝑙) is operator in 𝐿²(0,∞) with the kernel 𝑆_𝛾^(𝑙)𝑥−𝑥^′=2𝜋¹₋₁𝑃_𝑙(𝑡)𝑆_𝛾𝑥−𝑥^′;𝑡𝑑𝑡. (B.4) Now, comparing the definition (81) and (B.1) and applying the equality sinh𝑦𝜃=1sin𝜃sinh𝜋𝑦2𝜋^∞_−∞𝑒^{−𝑖𝑦𝑥}𝑑𝑥cosh𝑥+cos𝜃,0<𝜃<𝜋, (B.5) we see that 𝑆_𝛾𝑡;𝑦=^∞_−∞𝑒^{−𝑖𝑦𝑥}𝑆_𝛾(𝑥;𝑡)𝑑𝑥. (B.6) By (B.1) and (B.4), this yields 𝑆_𝛾^(𝑙)𝑦=^∞_−∞𝑒^{−𝑖𝑦𝑥}𝑆_𝛾^(𝑙)(𝑥)𝑑𝑥. (B.7)

One can check that the functions in (84) and (B.7) coincide.

Since the operator 𝐒_𝛾^(𝑙) is of convolution type, we conclude that it is unitarily equivalent to the operators 𝐒_𝛾^(𝑙) of multiplication by functions 𝑆_𝛾^(𝑙)(𝑦) in the space 𝐿²(ℝ¹). Therefore, 𝜎𝐒_𝛾^(𝑙)=𝜎𝐒_𝛾^(𝑙)=Ran𝑆_𝛾^(𝑙)𝑦. (B.8)

Recall that the number 𝛾^∗ is the unique solution of the equation 𝑏₁(𝛾)=1.

As it was mentioned in Lemma 19, for any 𝛾<𝛾^∗ there exists the number 𝜖_𝛾>0 such that for all 𝑙≥0sup_𝑦𝑆_𝛾^(𝑙)𝑦<1−𝜖_𝛾. (B.9)

The equality (B.3), (B.8), and (B.9) proved the inequality in the lemma.

Acknowledgments

This work was partially supported by the University Putra Malaysia under research university grant scheme (RUGS), and project code is 9190883. This work is also partially supported by the Fundamental Science Foundation of Uzbekistan. The authors are indebted to the anonymous referees for a number of constructive comments.