Number of bound states of the Schroedinger operator of a system of three bosons in an optical lattice

TL;DR

This paper analyzes the spectral properties of a three-boson system on a 1 or 2-dimensional lattice, establishing the finiteness of bound states, their exponential decay, and regularity with respect to momentum.

Contribution

It provides a detailed description of the essential spectrum and proves the finiteness and regularity of bound states for the lattice Schrödinger operator.

Findings

Finiteness of bound states below the essential spectrum.

Exponential decay of bound states at infinity.

Eigenvalues depend smoothly on the center of mass momentum.

Abstract

We consider the Hamiltonian $\hat {\mathrm{H}}_{\mu}$ of a system of three identical particles(bosons) on the $d-$ dimensional lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive potential $\mu<0$. We describe precise location and structure of the essential spectrum of the Schr\"odinger operator $H_\mu(K),K\in \T^d$ associated to $\hat {\mathrm{H}}_\mu$ and prove the finiteness of the number of bound states of $H_\mu(K),K\in \T^d$ lying below the bottom of the essential spectrum. Moreover, we show that bound states decay exponentially at infinity and eigenvalues and corresponding bound states of $H_\mu(K),K\in \T^d$ are regular as a function of center of mass quasi-momentum $K\in \T^d$.