Author： Breimesser S.V.
Publisher： Springer Publishing Company
ISSN： 1385-0172
Source： Mathematical Physics, Analysis and Geometry, Vol.6, Iss.1, 2003-02, pp. : 29-57
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Abstract
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.
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