Schr\"odinger operators with complex-valued potentials and no resonances

TL;DR

This paper constructs examples of complex-valued potentials for Schr"odinger operators in dimensions three and higher that have no resonances, revealing new spectral properties and behaviors of wave equations with such potentials.

Contribution

It provides explicit examples of nontrivial, compactly supported complex potentials with no resonances in dimensions three and higher, and analyzes their impact on wave equation solutions.

Findings

Existence of complex potentials with no resonances in $d eq 2$

Potential isophasality with the Laplacian

Super-exponential decay of wave solutions in fixed space variables

Abstract

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away from the origin. These Schr\"odinger operators are isophasal and have the same scattering phase as the Laplacian on $\Real^d$. In odd dimensions $d\geq 3$ we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is super-exponentially decaying in time.