Asymptotics of Eigenvalues of the Two-particle Schrödinger operators on lattices

Abstract

The Hamiltonian of a system of two quantum mechanical particles moving on the $d$-dimensional lattice ${\mathbb{Z}}^d$ and interacting via zero-range attractive pair potentials is considered. For the two-particle energy operator $H_{\mu }(K),$ $K\in \T^d=(-π,π]^d$ -- the two-particle quasi-momentum, the existence of a unique positive eigenvalue $z(\mu ,K)$ above the upper edge of the essential spectrum of $H_{\mu }(K)$ is proven and asymptotics for $z(\mu ,K)$ are found when $\mu$ approaches to some ${\mu }_0(K)$ and $K\to 0.$

References in corpus (1)

• Repulsively bound atom pairs in an optical lattice