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

[2404.18422] Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2404.18422 (math) [Submitted on 29 Apr 2024 ( v1 ), last revised 14 Apr 2026 (this version, v3)] Title: Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations Authors: Arup Chattopadhyay , Teun D.H. van Nuland , Chandan Pradhan View a PDF of the paper titled Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations, by Arup Chattopadhyay and 2 other authors View PDF HTML (experimental) 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 $$\frac{d^n}{dt^n}f(H+tV)|_{t=0}$$ 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 $$f(H+V)=\sum_{n=0}^\infty\frac{1}{n!}\frac{d^n}{dt^n} f(H+tV)\big|_{t=0}$$ exists and converges absolutely in operator norm. Finally, assuming that $V(H-i)^{-p}\in\mathcal{S}^{s/p}$ for $p=1,\ldots,s$ 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 $\eta_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)\eta_{k,H,V}(x)dx,$$ exist for every $k=1,2,3,\ldots$, 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. Comments: Implemented referee's comments. Appearing in the Journal of Functional Analysis Subjects: Functional Analysis (math.FA) MSC classes: 47A55, 46A56, 47B93, 47A60, 58B34 Cite as: arXiv:2404.18422 [math.FA] (or arXiv:2404.18422v3 [math.FA] for this version) https://doi.org/10.48550/arXiv.2404.18422 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chandan Pradhan [ view email ] [v1] Mon, 29 Apr 2024 04:41:16 UTC (22 KB) [v2] Tue, 19 Nov 2024 00:19:16 UTC (46 KB) [v3] Tue, 14 Apr 2026 18:40:29 UTC (46 KB) Full-text links: Access Paper: View a PDF of the paper titled Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations, by Arup Chattopadhyay and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2024-04 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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