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

We consider the Schrödinger operator $H\left(\mu \right)={\nabla }_A^{⁎}{\nabla }_A+\mu V$ on a Riemannian manifold M of bounded geometry, where $\mu >0$ is a coupling parameter, the magnetic field $B=dA$ 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 μ, the spectrum of $H\left(\mu \right)$ in the interval $[0,E_0\mu ]$ has a gap, the spectral projection of $H\left(\mu \right)$, corresponding to the interval $(-\infty ,\lambda ]$ with λ in the gap, belongs to the Roe $C^{⁎}$-algebra $C^{⁎}\left(M\right)$ of the manifold M, and, if M is not compact, its class in the K theory of $C^{⁎}\left(M\right)$ is trivial.