Non-self-adjoint operators, infinite determinants, and some applications

TL;DR

This paper explores spectral properties of non-self-adjoint operators, develops new determinant formulas, and connects these to scattering theory and spectral shift functions in multiple dimensions.

Contribution

It introduces a novel analysis of non-self-adjoint perturbations, extending classical formulas and linking determinants with spectral shift functions in higher dimensions.

Findings

Derived local and global Weinstein-Aronszajn formulas.

Connected perturbation determinants with Krein's spectral shift function.

Extended Jost-Pais formula to multi-dimensional operators.

Abstract

We study various spectral theoretic aspects of non-self-adjoint operators. Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of a variant of the Birman-Schwinger principle as well as local and global Weinstein-Aronszajn formulas. Our applications include a study of suitably symmetrized (modified) perturbation determinants of Schr\"odinger operators in dimensions n=1,2,3 and their connection with Krein's spectral shift function in two- and three-dimensional scattering theory. Moreover, we study an appropriate multi-dimensional analog of the celebrated formula by Jost and Pais that identifies Jost functions with suitable Fredholm (perturbation) determinants and hence reduces the latter to simple Wronski determinants.