The Faddeev Equation and the Essential Spectrum of a Model Operator Associated with the Hamiltonian of a Nonconserved Number of Particles

Amodel describing a truncated operatorH (truncated with respect to the number of particles) and acting in the direct sum of zero-, one-, and two-particle subspaces of fermionic Fock space F a (L2 (T3)) over L2(T3) is investigated. The location of the essential spectrum of the model operator H is described by means of the spectrum of the Friedreich model h(p); p ∈ T. Moreover, for the resolvent of H, the Faddeev type system of integral equations is obtained.


Introduction
The main goal of the present paper is to give a thorough mathematical treatment of some properties for a model operator  with emphasis on the essential spectrum and its resolvent.This operator, associated to a system describing two identical fermions and one particle of another nature in interactions, without conservation of the number of particles on the three-dimensional lattice, acts in the direct sum of zero-, one-, and two-particle subspaces of fermionic Fock space F  ( 2 (T 3 )) over  2 (T 3 ).
In statistical physics Minlos and Spohn [1], Malyshev and Minlos [2], solid-state physics Mattis [3], Mogilner [4] and the theory of quantum fields Friedrichs [5], Buhler et al. [6] some important problems arise where the number of quasiparticles is bounded, but not fixed.Sigal et al. [7] 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  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 () of the Fock space, consisting of  ≤  particles [1,[4][5][6][7][8].
In the works [9][10][11][12][13][14][15][16], the location of essential spectra of model operators, associated with a system describing three particles, without conservation of the number of particles, was investigated.However, the corresponding operators act in the direct sum of zero-, one-, and two-particle subspaces of Fock space or bosonic Fock space, in some cases (see, e.g., [9,10]) over  2 (T 3 ).We also refer to [17] for essential spectrum of discrete Schrödinger operators, associated to a system of two identical fermions and one particle of another nature in interactions.
Recently, the authors in [18,19] have proved that the fermionic Fock space case has some assertions being related to the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model (FM) has a threshold resonance).These results show that this effect does not hold even this FM has a threshold resonance while it holds in the (bosonic) Fock space case.This fact pushes us to study the resolvent and essential spectrum of the investigating operator.
In the present paper, under some smoothness assumptions we obtain the location of the essential spectrum of the model operator , described by means of the spectrum 2 Advances in Mathematical Physics of the Friedreich model ℎ(),  ∈ T 3 , and we derive the Faddeev type system of integral equations for the components of the resolvent of this model operator and find the form for resolvent.
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  in H (3) .Some spectral properties of the corresponding channel operator and Friedrichs models (),  ∈ T 3 , are given in Section 3. Section 4 deals with the review of the Faddeev type system of integral equations for the eigenfunction of operator .In Section 5, we represent the main results (Theorems 9 and 11) and the sketch of their proofs.

Conventions and Definition of the Model Operator
Let T 3 be the three-dimensional torus, the cube (−, ] 3 with appropriately identified sides.We remark that the torus T 3 will always be considered as an abelian group with respect to the addition and multiplication by the real numbers regarded as operations on R 3 modulo (2Z) 3 .Denote by  2 as ((T 3 ) 2 ) the subspace of antisymmetric functions of the Hilbert space  2 ((T 3 ) 2 ).
For  ∈  2 (T 3 ) we define the following operator: ( Let   be annihilation (creation) operators [5] defined in the Fock space for  <  ( > ).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 being an adjoint of the annihilation operator is an operator that increases the number of particles in a given state by one.
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   ≡ 0 for all | − | > 1.So, a model describing a truncated operator  acts in the Hilbert space H (3) the operators  00 ,  0 11 , and  0 22 are multiplication by the functions  0 , (⋅), and (⋅, ⋅) in H 0 , H 1 , and H 2 , respectively.
Remark that under these conditions the operator  is bounded and self-adjoint.

The Spectrum of the Channel Operator and of the Friedrichs Model
To study the essential spectrum, along with the operator , we also consider a bounded self-adjoint operator  ch acting in Ĥ =  2 (T 3 ) ⊕  2 ((T 3 ) 2 ) with form This operator has a characteristic property of a channel operator (see, e.g., [20]) of three-particle discrete Schrödinger operator.Therefore, we call it a channel operator, corresponding to the model operator .Note that the channel operator  ch has a simpler structure than the , and therefore the study of the spectral properties of  ch plays an important role in future studies of the spectrum of .
Since  ch commutes with the abelian group of multiplication operators   , by the function (⋅): the decomposition of Ĥ into Ĥ = ∫ ∈T 3 H (2) , where Here (),  ∈ T 3 , is a Friedrichs model being bounded, self-adjoint, and acting in H (2) by the rule where and ℎ 0 (),  ∈ T 3 , is a multiplication operator by the function   (⋅) := (, ⋅): Remark 1.The spectral properties of such type of Friedrichs models are studied in [9,10].
According to the theorem on the spectrum of decomposable operators (see, e.g., [21,Theorem XIII.85]) from (7), we obtain the following.
Theorem 2. For the essential spectrum   () of the operator  the equality, holds, where   (()) is the discrete spectrum of the operator (),  ∈ T 3 , and 3.1.The Spectral Properties of the Friedrichs Model (),  ∈ T 3 .Since rank  ≤ 3, and then in accordance with the stability of the essential spectrum under finite rank perturbations, the essential spectrum  ess (ℎ()) of ℎ() coincides with the spectrum of ℎ 0 (), and, namely, the equality, holds, where () and () are defined by For any  ∈ T 3 we define an analytic function Δ(, ) (the Fredholm determinant associated with the operator that is, where  0 (; ),  ∈ C \ [(), ()], is the resolvent of ℎ 0 (),  ∈ T 3 .
Proof.The equation that is, the system of equations, is equivalent to the system, The solutions of ( 17) and ( 19) are connected with relations Since the determinant of ( 19) is equal to Δ(; ), the equation, () = ,  ∈ H (2) , has nontrivial solution if and only if Δ(; ) = 0.

The Faddeev Type Equation
where Clearly, according to Theorem 2 and Lemma 3 the equalities Σ = ( ch ) and  two = ∪ ∈T 3 in  2 (T 3 ).
One may check that the operator is a multiplication operator by the function Δ(⋅; ) in the space  2 (T 3 ), where
Clearly, (, ) is continuous matrix-valued function in T 3 , a fact making the boundedness of ().
Since every component of () is compact so is (), a fact that together boundedness of  −1 () ends the proof.
Next assertion establishes connection between eigenvalues of the operators  and T() =  −1 ()().Lemma 8.The number  ∈ C \ Σ is an eigenvalue of the operator  if and only if the number  = 1 is an eigenvalue of T().

Proof.
Necessity.Let  ∈ C \ Σ be an eigenvalue of  with a corresponding eigenfunction ; that is, a system has a nontrivial solution  = ( 0 ,  1 ,  2 ).

Formulation and Proof of the Main Results
The first main result of the paper is given in the following theorem.
Theorem 9.The essential spectrum   () of  coincides with the set Σ; that is, Proof.
Then from (49) we obtain Let   (),  ∈ N, be a characteristic function of a set and (  ) Lebesgue measure of   .A sequences of trial functions { () } is chosen by where Here the function   ∈  2 (T 3 ) is found from the orthogonality condition { () }; that is, since where the notation ,  admits function-value either  or .
In case  0 ∈ [ min ,  max ], we chose ( 0 ,  0 ) ∈ (T 3 ) 2 as  = ( 0 ,  0 ) and we can construct an appropriate sequence of trial functions as Setting where is the resolvent of the operator  0 22 , we obtain the resolvent equation In our case resolvent Equation (67) is not compact.Therefore to overcome this difficulty, we derive Faddeev type system of integral equations (see, e.g., [23]) for the components of the resolvent.

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We write (67) as the form that is, Defining the unitary operator  : of (69) into  12  2 (), we take where ) . Set According to X() = ()() we write equation for X() as where  is the identity operator in H (3) .By analytic continuation, this holds for any  ∉ Σ ∪ .Thus, for any such , the operator  −  has a bounded inverse.Therefore ()\Σ consists of isolated points and only the frontier points of Σ are possible their limit points.Finally, since () has finite rank residues at  ∈ , we conclude that () \ Σ belongs to the discrete spectrum   () of , which completes the proof of Theorem 9. Now we derive resolvent form (64) of the operator .