Geometrical Aspects of Spectral Theory and Value Distribution for Herglotz Functions
Geometrical Aspects of Spectral Theory and Value Distribution for Herglotz Functions
复制标题
谱理论的几何方面和 Herglotz 函数的值分布
DOI：
--
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
D. Pearson
中科院分区：
文献类型：
--
作者：
S. V. Breimesser;D. Pearson
在本文中，我们展示了如何谱理论的Herglotz功能和微分算子是相关的，并依赖于几何性质的复杂的上半平面，视为双曲空间。建立了勒贝格可测函数f：R→R的值分布理论，并引入了与任意给定的Herglotz函数F相联系的值分布函数.我们将赫格洛茨函数边界值的值分布理论与半直线上薛定谔方程解的渐近性描述联系起来。我们建立了两个结果，这两个结果在理解具有稀疏势的薛定谔算子的渐近值分布及其对谱理论的影响方面起着关键作用。
In this paper we show how spectral theory for Herglotz functions and differential operators is related to and dependent on the geometrical properties of the complex upper half-plane, viewed as a hyperbolic space. We establish a theory of value distribution for Lebesgue measurable functions f: R→R and introduce the value distribution function associated with any given Herglotz function F. We relate the theory of value distribution for boundary values of Herglotz functions to the description of asymptotics for solutions of the Schrödinger equation on the half-line. We establish two results which play a key role in understanding asymptotic value distribution for Schrödinger operators with sparse potentials, and its implications for spectral theory.