Bound states of moving potential wells in discrete wave mechanics

Dipartimento di Fisica, Politecnico di Milano - Piazza L. da Vinci 32, I-20133 Milano, Italy and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Sezione di Milano Piazza L. da Vinci 32, I-20133 Milano, Italy

Received: 14 November 2017
Accepted: 20 December 2017

Abstract

Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger 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 Schrödinger 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 Schrödinger 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.