Bound states of moving potential wells in discrete wave mechanics

TL;DR

This paper investigates how lattice discretization affects bound states in moving potential wells within discrete wave mechanics, revealing that lattice effects can increase bound states and induce resonance phenomena.

Contribution

It demonstrates that lattice discretization breaks Galilean invariance, leading to more bound states and resonance states for moving potential wells, unlike in continuous Schrödinger dynamics.

Findings

Lattice effects increase the number of bound states for moving wells.

Moving potential wells can exhibit quasi-bound (resonance) states on a lattice.

Galilean invariance is broken in discrete wave mechanics, affecting bound state properties.

Abstract

Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schr\"odinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schr\"odinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schr\"odinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity $v$. While in the continuous limit the number of bound states does not depend on the drift velocity $v$, as one expects from the covariance of ordinary Schr\"odinger equation for a Galilean boost, lattice effects can lead to a larger number of bound states for the moving potential well as compared to the potential well at rest. Moreover, for a moving potential bound states on a lattice become rather generally quasi-bound (resonance) states.