Monotonicity and Concavity Properties of The Spectral Shift Function
| dc.creator | Gesztesy, F. | |
| dc.creator | Makarov, K. A. | |
| dc.creator | Motovilov, A. K. | |
| dc.date | 1999-09-14 | |
| dc.date.accessioned | 2026-07-07T05:30:45Z | |
| dc.date.available | 2026-07-07T05:30:45Z | |
| dc.description | Let $H_0$ and $V(s)$ be self-adjoint, $V,V'$ continuously differentiable in trace norm with $V''(s)\geq 0$ for $s\in (s_1,s_2)$, and denote by $\{E_{H(s)}(λ)\}_{λ\in\bbR}$ the family of spectral projections of $H(s)=H_0+V(s)$. Then we prove for given $μ\in\bbR$, that $s\longmapsto \tr\big (V'(s)E_{H(s)}((-\infty, μ))\big) $ is a nonincreasing function with respect to $s$, extending a result of Birman and Solomyak. Moreover, denoting by $ζ(μ,s)=\int_{-\infty}^μdλξ(λ,H_0,H(s))$ the integrated spectral shift function for the pair $(H_0,H(s))$, we prove concavity of $ζ(μ,s)$ with respect to $s$, extending previous results by Geisler, Kostrykin, and Schrader. Our proofs employ operator-valued Herglotz functions and establish the latter as an effective tool in this context. | |
| dc.description | LaTeX, 15 pages | |
| dc.identifier | https://arxiv.org/abs/math/9909076 | |
| dc.identifier | http://arxiv.org/abs/math/9909076 | |
| dc.identifier | CMS Conf. Proc. Series (AMS, Providence, RI) {\bf 29} (2000), 207-222 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79097 | |
| dc.subject | Spectral Theory | |
| dc.title | Monotonicity and Concavity Properties of The Spectral Shift Function | |
| dc.type | text |