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.

26 pages; v2: final version

References in corpus (4)

• Gaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry
• Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in K-theory
• Delocalized spectra of Landau operators on helical surfaces
• On topological obstructions to the existence of non-periodic Wannier bases