Resonant rigidity for Schr\"odinger operators in even dimensions

TL;DR

This paper investigates the distribution of resonances for Schr"odinger operators in even-dimensional Euclidean spaces, establishing results on their density, inverse problems, and the determination of heat coefficients from resonances.

Contribution

It provides new results on the density of resonances, inverse spectral properties, and the compactness of isoresonant potential sets for Schr"odinger operators in even dimensions.

Findings

Infinitely many resonances for non-trivial potentials in dimensions not equal to 4.

Resonance sets determine heat coefficients for smooth potentials.

Isoresonant potentials form compact sets.

Abstract

This paper studies the resonances of Schr\"odinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schr\"odinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schr\"odinger operators $-\Delta +V_1$ and $-\Delta +V_2$ have the same poles, with both potentials bounded, compactly supported and real-valued, if k is a natural number and if $V_1\in H^k({\mathbb R}^d; {\mathbb R})$, then $V_2\in H^k$ as well. Moreover, we prove that certain sets of isoresonant potentials are compact. We also show that the poles of the resolvent for a smooth potential determine the heat coefficients and that the (resolvent) resonance sets of two bounded, real-valued potentials with compact support cannot differ by a nonzero finite number of elements away from $0$.