Bound States of a System of Two Fermions on Invariant Subspace

We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice 3  with special potential ˆ v . The corresponding Shrödinger operator ( ) H k of the system has an invariant subspace ( ) 3 123 L −  , where we study the eigenvalues and eigenfunctions of its restriction ( ) 123 H − k . Moreover, there are shown that ( ) 123 1 2 , , H k k π − has also infinitely many invariant subspaces ( ) 123 , n n − ∈  R , where the eigenvalues and eigenfunctions of eigenvalue problem ( ) ( ) 1 2 123 , , , H k k f zf f n π − = ∈ R


Introduction
The nature of bound states of two-particle cluster operators for small parameter values was first studied in detail by Minlos and Mamatov [1] and then in a more general setting by Minlos and Mogilner [2]. In [3], Howland  , where k is the total quasi-momentum of a system. Moreover, eigenfunctions of ( ) studied in the works [7] [9].
The discrete spectrum of the two-particle continuous Shrödinger operator h V λ λ = −∆ + was studied by many authors, with the conditions for the potential V formulated in its coordinate representation. The condition for the finiteness of the set of negative elements of the spectrum and the absence of positive eigenvalues of h λ can be found in [10]. If It is known that when the coupling constant λ decreases, the bound state energies of h λ tend to the boundary of the continuous spectrum (see [10]) and for some finite λ are on the boundary. Two questions then arise: Does a bound or virtual state correspond to such a threshold state (i.e., is the corresponding wave function square-integrable)? And where do the bound states "disappear to" as λ decreases further? The study of the first question was the subject in [11] [12]. Regarding the second question, it turns out that the bound state disappears by being absorbed into the continuous spectrum and becomes a resonance [5].
Here, we consider bound states of the Hamiltonian Ĥ (see (1)) of a system of two fermions on the three-dimensional lattice 3  with the special potential v (see (5)). In other words, we study the discrete spectrum of a family of the Shrödinger operators ( ) , , k k k = ∈  k , (see (3)) corresponding to Ĥ in the invariant subspace

Description of the Hamiltonian and Expansion in a Direct Integral
The free Hamiltonian 0 H of a system of two fermions on a three-dimensional lattice 3  usually corresponds to a bounded self-adjoint operator acting in the Hilbert space }   3  3  3  3  2  2   : : , , Here, m is the fermion mass, which we assume to be equal to unity in what follows, 1 I ∆ = ∆ ⊗ and 2 I ∆ = ⊗ ∆ , where I is the identity operator, and the lattice Laplacian ∆ is a difference operator that describes a translation of a particle from a side to a neighboring side, : .
x y x y Hereafter, we assume that : : The unperturbed operator From (3) and (4), it follows that 2  3  1  2  3  1  2  3  1  2  3 , , , The perturbation operator V is an integral operator in ( ) and belongs to the class of Hilbert-Schmidt operators 2 Σ .
In this work, we consider the operator The function : and belongs to ( ) 2   . The kernel v , of the integral operator V, i.e., the Fouri- The spectrum of V consists of the set the operator V is a Hilbert-Schmidt operator and is hence compact. By the Weyl theorem [10], the essential spectrum of ( ) H k coincides with the spectrum of , then the spectrum of consists of eigenvalues of the form ( ) 6 , v n n − ∈ and the essential spectrum is { } , then there exists a potential v such that ( ) H k has an infinite number of eigenvalues outside the continuous spectrum (see [4] [14]).
We recall some notations and known facts. For any self-adjoint operator B acting in a Hilbert space H without an essential spectrum to the right of µ ∈  , we let ( ) , The following proposition (the Birman-Schwinger principle) holds [9]. holds.

Vf p p p v p s p s p s f s s s s s s
which completes the proof of the lemma.
We denote by ( ) . Formula (11) shows that the restriction where I is the identity operator and , , is a two-dimensional two-particle operator acting in ( ) 2 12 L −  by i.e. the three-dimensional problem reduces to the two-dimensional problem.

Eigenvalues of the Operator
It is known that the essential spectrum of ( ) ( ) Here C λ is the normalizing multiplicity.
The following lemma establishes a connection between the operators