Isoresonant complex-valued potentials and symmetries

TL;DR

This paper constructs complex-valued potentials on symmetric Riemannian manifolds that preserve the resonance structure of the free Laplacian, revealing deep connections between symmetries and spectral properties.

Contribution

It introduces a method to create complex potentials that maintain the resonance spectrum of the Laplacian on symmetric manifolds.

Findings

Resonance poles are preserved under specific complex-valued potentials.

Construction of potentials relies on manifold symmetries.

Resonance multiplicities remain unchanged with the new potentials.

Abstract

Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(\Delta-z)^{-1}, z\in\C\setminus\R^{+},$ has a meromorphic continuation through $\R^{+}$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $\Delta+V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian.