The Number and Location of Eigenvalues of the Two Particle Discrete Schrödinger Operators

On the spectrum of the two-particle Schrödinger operator with point potential: one dimensional case

“…In the discrete case, there were also found conditions for the existence of the eigenvalues as well as their numbers for the Hamiltonian of the system of two particles depending on parameters. For example, in [12][13][14][15], the Hamiltonian h of the system of two quantum particles moving on a one and three-dimensional lattices interacting via some attractive potential was considered. Conditions for the existence of eigenvalues of the two-particle Schrödinger operator h µ (k), k ∈ T, d = 1, 3; µ, associated with the Hamiltonian h, were studied depending on the energy of the particle interaction µ and total quasi-momentum k ∈ T d .…”

On eigenvalues and virtual levels of a two-particle Hamiltonian on a d-dimensional lattice

“…The spectral properties of this operator h(k) for the one dimensional case were studied in [15] and more general case in [16]. For general case ε(p) satisfying some conditions and v(p − s) = µ 0 + Hamiltonian ĥµλ , µ, λ 0, describing the motion of one quantum particle on a three-dimensional lattice in an external field and more general case were investigated in the papers [18] and [19], respectively. In [20] a class of potentials is found for which the discrete spectrum of the two-particle Schrödinger operator h(k) is preserved when h(k) is perturbed by a potential from this class.…”