Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential

Abstract

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, ${\mathcal{H}}_{\hslash ,V}:=-{\hslash }^{-2}{\mathcal{L}}_{\hslash }+V$ on the lattice $\hslash {\mathbb{Z}}^n,$ where ${\mathcal{L}}_{\hslash }$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that ${\mathcal{H}}_{\hslash ,V}$ 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 $\hslash \to 0$.

25 pages