A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908510884724736 |
|---|---|
| author | Kordyukov, Yuri A. Manuilov, Vladimir M. |
| author_facet | Kordyukov, Yuri A. Manuilov, Vladimir M. |
| contents | We consider the Schrödinger operator $H(μ) = \nabla_{\bf A}^*\nabla_{\bf A} + μV$ on a Riemannian manifold $M$ of bounded geometry, where $μ>0$ is a coupling parameter, the magnetic field ${\bf B}=d{\bf A}$ and the electric potential $V$ are uniformly $C^\infty$-bounded, $V\geq 0$. We assume that, for some $E_0>0$, each connected component of the sublevel set $\{V<E_0\}$ of the potential $V$ is relatively compact. Under some assumptions on geometric and spectral properties of the connected components, we show that, for sufficiently large $μ$, the spectrum of $H(μ)$ in the interval $[0,E_0μ]$ has a gap, the spectral projection of $H(μ)$, corresponding to the interval $(-\infty,λ]$ with $λ$ in the gap, belongs to the Roe $C^*$-algebra $C^*(M)$ of the manifold $M$, and, if $M$ is not compact, its class in the $K$ theory of $C^*(M)$ is trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17746 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator Kordyukov, Yuri A. Manuilov, Vladimir M. Differential Geometry Mathematical Physics Operator Algebras Spectral Theory We consider the Schrödinger operator $H(μ) = \nabla_{\bf A}^*\nabla_{\bf A} + μV$ on a Riemannian manifold $M$ of bounded geometry, where $μ>0$ is a coupling parameter, the magnetic field ${\bf B}=d{\bf A}$ and the electric potential $V$ are uniformly $C^\infty$-bounded, $V\geq 0$. We assume that, for some $E_0>0$, each connected component of the sublevel set $\{V<E_0\}$ of the potential $V$ is relatively compact. Under some assumptions on geometric and spectral properties of the connected components, we show that, for sufficiently large $μ$, the spectrum of $H(μ)$ in the interval $[0,E_0μ]$ has a gap, the spectral projection of $H(μ)$, corresponding to the interval $(-\infty,λ]$ with $λ$ in the gap, belongs to the Roe $C^*$-algebra $C^*(M)$ of the manifold $M$, and, if $M$ is not compact, its class in the $K$ theory of $C^*(M)$ is trivial. |
| title | A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator |
| topic | Differential Geometry Mathematical Physics Operator Algebras Spectral Theory |
| url | https://arxiv.org/abs/2412.17746 |