Discrete time-dependent wave equation for the Schr\"{o}dinger operator with unbounded potential

TL;DR

This paper studies the semiclassical wave equation for the discrete Schrödinger operator with unbounded potential, proving spectral discreteness, well-posedness, and convergence of solutions as the semiclassical parameter tends to zero.

Contribution

It establishes spectral properties, well-posedness, and solution limits for the discrete Schrödinger wave equation with unbounded potentials.

Findings

Discrete Schrödinger operator has purely discrete spectrum under unbounded potential.

Cauchy problem is well-posed with regular coefficients and weakly well-posed with distributional coefficients.

Classical and very weak solutions are recovered as semiclassical parameter approaches zero.

Abstract

In this article, we investigate the semiclassical version of the wave equation for the discrete Schr\"{o}dinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}_{\hbar,V}$ has a purely discrete spectrum when the potential $V$ satisfies the condition $|V(k)|\to \infty$ as $|k|\to\infty$. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter $\hbar\to 0$.