Norm-ideal perturbations of one-parameter semigroups and applications

TL;DR

This paper develops a new framework for perturbing generators of one-parameter semigroups using norm-ideal perturbations, leading to refined equivalence relations and improved convergence properties, with applications to non-self-adjoint Schrödinger operators.

Contribution

It introduces a graded family of generator equivalence relations based on Schatten-von Neumann ideals, enhancing the classical perturbation theory for semigroups.

Findings

Refined convergence properties for Dyson series tail

Framework applicable to non-self-adjoint Schrödinger operators

Stronger-than-expected perturbation convergence results

Abstract

The notion of equivalence classes of generators of one-parameter semigroups based on the convergence of the Dyson expansion can be traced back to the seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of their Functional Analysis monograph, developed the theory in minute detail. Following their approach of regarding the Dyson expansion as a central object, in the first part of this paper we examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces. This allows us to develop a graded family of equivalence relations on generators, which refine the classical notion and provide stronger-than-expected properties of convergence for the tail of the perturbation series. We then show how this framework realises in the context of non-self-adjoint Schrodinger operators.