Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential (2306.02409v1)

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 $\hbar\to 0$.