Semiclassical Resonances of Schr\"odinger operators as zeroes of regularized determinants

TL;DR

This paper demonstrates that resonances of semiclassical Schrödinger operators with long-range perturbations are characterized as zeros of specific regularized determinants, providing factorizations, bounds, and spectral shift representations.

Contribution

It establishes a link between resonances and zeros of regularized determinants for long-range perturbations, including explicit factorizations and semiclassical bounds.

Findings

Resonances are zeros of natural perturbation determinants.

Factorizations of determinants involve products over resonances and exponential factors.

Provides semiclassical bounds on derivatives of the phase function and a spectral shift function representation.

Abstract

We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \prod_{w = {\rm resonances}}(z-w) \exp (\varphi_p(z,h)) $ and give semiclassical bounds on $ \partial_z \varphi_p $ as well as a representation of Koplienko's regularized spectral shift function. Here the index $ p \geq 1 $ depends on the decay rate at infinity of the perturbation.