Lattice oscillator model, scattering theory and a many-body problem

TL;DR

This paper introduces a supersymmetric lattice model for the quantum harmonic oscillator, enabling analytical and numerical analysis of its spectrum and applications to scattering and many-body problems in one dimension.

Contribution

It presents a novel supersymmetric lattice oscillator model that simplifies the analysis of spectra, scattering lengths, and many-body ground states.

Findings

Ground state annihilated by the defined operator

Numerical excitation spectrum obtained

Exact many-body ground state calculated

Abstract

We propose a model for the quantum harmonic oscillator on a discrete lattice which can be written in supersymmetric form, in contrast with the more direct discretization of the harmonic oscillator. Its ground state is easily found to be annihilated by the annihilation operator defined here, and its excitation spectrum is obtained numerically. The versatility of the model is then used to calculate, in a simple way, the generalized position-dependent scattering length for a particle colliding with a single static impurity in a periodic potential and the exact ground state of an interacting many-body problem in a one-dimensional ring.