Higher-order spectral shift function for resolvent comparable perturbations

TL;DR

This paper develops higher-order trace formulas and spectral shift functions for self-adjoint operators with Schatten-von Neumann resolvent differences, extending previous results to broader classes of perturbations and functions.

Contribution

It generalizes existing trace formulas and spectral shift functions to more general Schatten-von Neumann perturbations, improving upon earlier restrictive conditions.

Findings

Established higher-order trace formulas for broad classes of functions.

Proved existence of real-valued spectral shift functions with near-uniqueness.

Extended the framework to encompass general Schatten-von Neumann resolvent differences.

Abstract

Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations $V$ and settles earlier attempts to encompass general perturbations with Schatten-von Neumman difference of resolvents, which led to more complicated trace formulas for more restrictive sets of functions $f$ and to analogs of spectral shift functions lacking real-valuedness and/or expected degree of uniqueness. Our proof builds on a general change of variables method derived in this paper and significantly refining those appearing in \cite{vNS21,PSS15,S17} with respect to several parameters at once.