Asymptotic Value Distribution for Solutions of the Schrödinger Equation

Abstract

We consider the Dirichlet Schrödinger operator T=−(d²/d x ²)+V, acting in L ²(0,∞), where Vis an arbitrary locally integrable potential which gives rise to absolutely continuous spectrum. Without any other restrictive assumptions on the potential V, the description of asymptotics for solutions of the Schrödinger equation is carried out within the context of the theory of value distribution for boundary values of analytic functions. The large xasymptotic behaviour of the solution v(x,λ) of the equation T f(x,λ)=λf(x,λ), for λ in the support of the absolutely continuous part μ_a.c. of the spectral measure μ, is linked to the spectral properties of this measure which are determined by the boundary value of the Weyl–Titchmarsh m-function. Our main result (Theorem 1) shows that the value distribution for v′(N,λ)/v(N,λ) approaches the associated value distribution of the Herglotz function m ^N(z) in the limit N→∞, where m ^N(z) is the Weyl–Titchmarsh m-function for the Schrödinger operator −(d²/d x ²)+Vacting in L ²(N,∞), with Dirichlet boundary condition at x=N. We will relate the analysis of spectral asymptotics for the absolutely continuous component of Schrödinger operators to geometrical properties of the upper half-plane, viewed as a hyperbolic space.