Stokes’ phenomenon and the absolutely continuous spectrum of one-dimensional Schrödinger operators
| Autor: | D. J. Gilbert, Alastair Wood |
|---|---|
| Rok vydání: | 2004 |
| Předmět: |
Pure mathematics
Differential equation Applied Mathematics One-dimensional Schrödinger operators Spectrum (functional analysis) Mathematical analysis Boundary (topology) Liouville-Green approximation One-dimensional Schrodinger operators Stokes’ phenomena Computational Mathematics Real-valued function Airy function Absolutely continuous spectrum Boundary value problem Complex plane Mathematics Self-adjoint operator |
| Zdroj: | Articles |
| ISSN: | 0377-0427 |
| DOI: | 10.1016/j.cam.2004.01.012 |
| Popis: | It is well known that the Airy functions, Ai(-x-µ) and Bi(-x-µ), form a fundamental set of solutions for the differential equation Lu(x):=-u''(x)-xu(x)=µu(x), 0 ≤ x < ∞, µ ∈ R, and that the spectrum of the associated selfadjoint operator consists of the whole real axis and is purely absolutely continuous for any choice of boundary condition at x=0. Also widely known is the fact that the semi-axis [-µ,∞) is an anti-Stokes' line for solutions of the differential equation Lu(z)=µu(z),z ∈ C, for each fixed value of the spectral parameter µ. In this paper, we show that this connection between the existence of anti-Stokes' lines on the real axis and points of the absolutely continuous spectrum holds under much more general circumstances. Further correlations, relating the Stokes' phenomenon to subordinacy properties of solutions of Lu=µu at infinity and to the boundary behaviour of the Titchmarsh-Weyl m-function on the real axis, are also deduced. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect