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

对于薛定谔算子， 光谱考虑包括共振的分布 威尔斯和控制的伪黎曼度量的雅可比 还有Agmon。 波动方程的有效几何是 由障碍物的形状和边界决定 条件 双曲流形的有效几何是 由庞加莱度量确定。 在每一种情况下， 研究将检查光谱数据如何反映捕获 或非捕获性质的潜力或能力的 度量来捕获测地线。 本研究的一个目标是 表示几何性质的迹公式的推导 在光谱数据方面的流形。 这项研究解决了相互作用的一般问题， 非紧流形的几何和谱性质。 首先，流形的几何如何决定谱 类型 第二，光谱数据如何对 流形的几何学和拓扑学。 三个应用案例 是电场有序介质的量子力学， 散射的波动方程，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.