Non-harmonic analysis of the wave equation for Schr\"{o}dinger operators with complex potential

TL;DR

This paper studies the wave equation for Schrödinger operators with complex potentials, establishing spectral properties and well-posedness results in various function spaces, including Sobolev and Gevrey spaces, even with singular coefficients.

Contribution

It provides new spectral analysis and well-posedness results for Schrödinger operators with complex potentials, extending understanding to cases with distributional singularities.

Findings

Operator has purely discrete spectrum under certain conditions

Cauchy problem is well-posed in Sobolev and Gevrey spaces

Well-posedness extends to cases with distributional singularities

Abstract

This article investigates the wave equation for the Schr\"{o}dinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-\Delta+V$, where $\Delta$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator $\mathcal{H}_0$, with $\operatorname{Re}(V)\geq 0$ and $\operatorname{Re}(V)(x)\to\infty$ 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 H\"{o}lder 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.