Lattice two-body problem with arbitrary finite range interactions

TL;DR

This paper provides an exact and efficient method to solve the two-body problem on a one-dimensional lattice with finite-range interactions, including bound, scattering, and low-energy states, with potential for generalizations.

Contribution

It introduces a polynomial-root based approach for characterizing bound states and a numerical method for locating low-energy resonances in lattice two-body problems.

Findings

All bound states correspond to roots of a polynomial with degree proportional to interaction range.

The method efficiently computes the full spectrum, including bound and scattering states.

Connections between the number of bound states and scattering lengths are established.

Abstract

We study the exact solution of the two-body problem on a tight-binding one-dimensional lattice, with pairwise interaction potentials which have an arbitrary but finite range. We show how to obtain the full spectrum, the bound and scattering states and the "low-energy" solutions by very efficient and easy-to-implement numerical means. All bound states are proven to be characterized by roots of a polynomial whose degree depends linearly on the range of the potential, and we discuss the connections between the number of bound states and the scattering lengths. "Low-energy" resonances can be located with great precission with the methods we introduce. Further generalizations to include more exotic interactions are also discussed.