Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential

Abstract

This article investigates the wave equation for the Schrödinger operator on ${\mathbb{R}}^n$, denoted as ${\mathcal{H}}_0:=-\hat{I}”+V$, where $\hat{I}”$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator ${\mathcal{H}}_0$, with $\mathrm{Re}(V)\ge 0$ and $\mathrm{Re}(V)(x)\to \infty$ as $|x|\to \infty$, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is Hölder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity.

25 pages