Perturbations and operator trace functions

TL;DR

This paper analyzes spectral trace functionals of the form tr f(D+A) for self-adjoint operators, deriving a Taylor expansion involving Gâteaux derivatives, extending previous finite-dimensional results to infinite dimensions with applications in noncommutative geometry and physics.

Contribution

It introduces a generalized Taylor expansion for spectral functionals in infinite-dimensional settings, involving divided differences and eigenbasis coefficients.

Findings

Derived a Taylor expansion for spectral functionals tr f(D+A).

Extended finite-dimensional results to infinite-dimensional operators.

Applicable to noncommutative geometry and physical models.

Abstract

We study the spectral functional tr f(D+A) for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of G\^ateaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives.