Mathematical Sciences: Relations Between Spectra and Geometry for Schrodinger Operators and Hyperbolic Manifolds

对于薛定谔算符，用于谱考虑的有效几何由共振势垒的分布组成，并且由Jacobi和Agmon的伪黎曼度规控制。波动方程的有效几何形状由障碍物的形状和边界条件决定。双曲流形的有效几何由Poincare度规决定。在每一种情况下，这项研究都将检查光谱数据如何反映势能的圈闭或非圈闭性质，或度量的圈闭测地线的能力。这项研究的一个目标是推导出用谱数据来表示流形的几何性质的迹公式。这项研究解决了非紧流形的几何性质和谱性质相互作用的一般问题。首先，流形的几何形状如何决定光谱类型。第二，光谱数据如何对流形的几何和拓扑施加约束。应用的三个例子是具有电场的有序介质的量子力学、波动方程的散射和无限体积双曲流形。

For the Schrodinger operator, the effective geometry for spectral considerations consists of the distribution of resonant wells and is controlled by the pseudo-Riemannian metric of Jacobi and Agmon. The effective geometry for the wave equation is determined by the shape of the obstacle and the boundary conditions. The effective geometry for the hyperbolic manifold is determined by the Poincare metric. In each of these cases, this research will examine how the spectral data reflects the trapping or non-trapping property of the potential or the capacity of the metric to trap geodesics. A goal of this research is the derivation of the trace formulas which express geometric properties of the manifold in terms of the spectral data. This research addresses the general question of the interplay of the geometric and spectral properties of non-compact manifolds. First, how does the geometry of the manifold determine the spectral types. Second, how does the spectral data place constraints on the geometry and topology of the manifold. Three cases of application are the quantum mechanics of ordered media with electric fields, scattering for the wave equation, and infinite volume hyperbolic manifolds.