标题： 在非周期性磁 Schrödinger 算子的谱投影的$K$理论中的一个消失定理 显示英文标题

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

摘要： 我们考虑在有界几何的黎曼流形$M$上的薛定谔算子$H(\mu) = \nabla_{\bf A}^*\nabla_{\bf A} + \mu V$，其中$\mu>0$是耦合参数，磁场${\bf B}=d{\bf A}$和电势$V$是一致$C^\infty$-有界的，$V\geq 0$。 我们假设，对于某个$E_0>0$，势函数$V$的子水平集$\{V<E_0\}$的每个连通分支是相对紧的。 在关于连通分支的几何和谱性质的一些假设下，我们证明了，当$\mu$足够大时，$H(\mu)$在区间$[0,E_0\mu]$中的谱有一个间隙，对应于区间$(-\infty,\lambda]$的$H(\mu)$的谱投影，其中$\lambda$在间隙中，属于流形$M$的 Roe$C^*$-代数$C^*(M)$，并且其在$C^*(M)$的$K$理论中的类是平凡的。 显示英文摘要

摘要： 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 its class in the $K$ theory of $C^*(M)$ is trivial.