The Exact Number of Eigenvalues of the Discrete Schrödinger Operators in One-Dimensional Case

We study two-particle Schrödinger operators $${H}_{\lambda\mu}(k)$$ , with the fixed quasi-momentum of particles pair $$k\in\mathbb{T}$$ , on $$L^{2,o}(\mathbb{T},\eta)$$ . These operators are associated to the Bose–Hubbard Hamiltonian $$\widehat{\mathbb{H}}_{\lambda\mu}$$ of a system of two identical quantum-mechanical particles (fermions) interacting via zero-range potential $$\mu\in\mathbb{R}$$ on one site and potential $$\lambda\in\mathbb{R}$$ on neighboring sites. We establish a partition of the $$\lambda-\mu$$ parameter-plane into several connected components where the Schrödinger operator $$H_{\lambda\mu}(k)$$ can have only a definite (constant) number of eigenvalues. The eigenvalues may locate and below the bottom of the essential spectrum and above its top.

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