A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator

Abstract

We consider the Schrödinger operator $H(\mu )={\nabla }_{\mathbf{A}}^{\ast }{\nabla }_{\mathbf{A}}+\mu V$ on a Riemannian manifold $M$ of bounded geometry, where $\mu >0$ is a coupling parameter, the magnetic field $\mathbf{B}=d\mathbf{A}$ and the electric potential $V$ are uniformly $C^{\infty }$-bounded, $V\ge 0$. 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 $\mu$, the spectrum of $H(\mu )$ in the interval $[0,E_0\mu ]$ has a gap, the spectral projection of $H(\mu )$, corresponding to the interval $(-\infty ,\lambda ]$ with $\lambda$ in the gap, belongs to the Roe $C^{\ast }$-algebra $C^{\ast }(M)$ of the manifold $M$, and, if $M$ is not compact, its class in the $K$ theory of $C^{\ast }(M)$ is trivial.

arXiv comment26 pages; v2: final version

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