Bound states of moving potential wells in discrete wave mechanics

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.