A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schr\"odinger operator

Abstract

We consider the Schr\"odinger operator $H(\mu) = \nabla_{\bf A}^*\nabla_{\bf A} + \mu V$ on a Riemannian manifold $M$ of bounded geometry, where $\mu>0$ 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 $\mu$, 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 $C^*$-algebra $C^*(M)$ of the manifold $M$, and, if $M$ is not compact, its class in the $K$ theory of $C^*(M)$ is trivial.