Asymptotics of Eigenvalues of the Two-particle Schr\"{o}dinger operators on lattices

TL;DR

This paper investigates the asymptotic behavior of eigenvalues of two-particle Schrödinger operators on lattices, establishing existence and asymptotics of bound states as interaction strength and quasi-momentum vary.

Contribution

It proves the existence of a unique positive eigenvalue above the essential spectrum and derives its asymptotics near critical interaction strength and zero quasi-momentum.

Findings

Existence of a unique positive eigenvalue above the essential spectrum.

Asymptotic formulas for eigenvalues as interaction strength approaches critical value.

Eigenvalue behavior as quasi-momentum approaches zero.

Abstract

The Hamiltonian of a system of two quantum mechanical particles moving on the $d$-dimensional lattice $\Z^d$ and interacting via zero-range attractive pair potentials is considered. For the two-particle energy operator $H_{\mu}(K),$ $K\in \T^d=(-\pi,\pi]^d$ -- 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 $K\to 0.$