Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice

Abstract

We consider the Hamiltonian of a system of two fermions on a two-dimensional lattice ${\mathbb{Z}}^2$ with a certain type potential. It is proved that the subspace of odd functions $L^o_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$ is represented as a direct sum of the subspaces $L^{eo}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$ and $L^{oe}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$, which are invariant under the operator $H(\mathbf {k}) {,\ }\mathbf {k}=(k_1, k_2) \in {\mathbb{T}}^2,$ associated with this Hamiltonian. For any $k_1 \in {({-\pi,}\pi ]}$, it is proved that the operator $H^{eo}(k_1,\pi)=H(k_1,\pi)\big|_{L^{eo}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}}$ has an infinite number of eigenvalues and for any $k_1 \in ({-\pi,}\pi {)}$, the operator $H^{oe}(k_1,\pi)=H(k_1,\pi)\big|_{L^{oe}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}}$ has a finite number eigenvalues lying to the left of the essential spectrum. An asymptotic formula is obtained for the number of eigenvalues of the operator $H^{oe}(k_1,\pi)$ as $k_1\rightarrow \pi.$