Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911593230499840 |
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| author | Chattopadhyay, Arup van Nuland, Teun D. H. Pradhan, Chandan |
| author_facet | Chattopadhyay, Arup van Nuland, Teun D. H. Pradhan, Chandan |
| contents | 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 $η_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)η_{k,H,V}(x)dx,$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18422 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations Chattopadhyay, Arup van Nuland, Teun D. H. Pradhan, Chandan Functional Analysis 47A55, 46A56, 47B93, 47A60, 58B34 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 $η_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)η_{k,H,V}(x)dx,$$ 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. |
| title | Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations |
| topic | Functional Analysis 47A55, 46A56, 47B93, 47A60, 58B34 |
| url | https://arxiv.org/abs/2404.18422 |