Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations

Abstract

Given $H$ self-adjoint, $V$ symmetric and relatively $H$-bounded, and $f:\mathbb{R}\to \mathbb{C}$ satisfying mild conditions, we show that the Gateaux derivative $$\frac{d^n}{d{t}^n}f(H+tV){|}_{t=0}$$ exists in the operator norm topology, for every natural $n$, give a new explicit formula for this derivative in terms of multiple operator integrals, and establish useful perturbation formulas for multiple operator integrals under relatively bounded perturbations. Moreover, if the $H$-bound of $V$ is less than 1, we obtain sufficient conditions on $f$ which ensure that the Taylor expansion $$f(H+V)=\sum _{n=0}^{\infty }\frac{1}{n!}\frac{d^n}{d{t}^n}f(H+tV){|}_{t=0}$$ exists and converges absolutely in operator norm. Finally, assuming that $V(H-i{)}^{-p}\in {\mathcal{S}}^{s/p}$ for $p=1,\dots ,s$ for some $s\in \mathbb{N}$ (for instance, when $H$ is an order 1 differential operator on an $s-1$ dimensional space), we show that the Krein--Koplienko spectral shift functions $\hat{I}{\cdot }_{k,H,V}$, satisfying $$Tr\left(f(H+V)-\sum _{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{d{t}^m}f(H+tV){|}_{t=0}\right)={\int }_{\mathbb{R}}{f}^{(k)}(x)\hat{I}{\cdot }_{k,H,V}(x)dx,$$ exist for every $k=1,2,3,\dots$, independently of $s$. The latter result (which is significantly stronger than \cite{vNS22}) is completely new also in the case that $V$ is bounded. The proof is based on \cite{PSS}, combined with a generalisation of the multiple operator integral compatible with \cite{HMvN}. We discuss applications of our results to quantum physics and noncommutative geometry.

Implemented referee's comments. Appearing in the Journal of Functional Analysis