Stokes Phenomenon and the Absolutely Continuous Spectrum of One-Dimensional Schrodinger Operators
Document Type
Article
Rights
This item is available under a Creative Commons License for non-commercial use only
Disciplines
1.1 MATHEMATICS
Abstract
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 ≔ 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 , 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.
DOI
https://doi.org/10.1016/j.cam.2004.01.012
Recommended Citation
D.J.Gilbert, D.J., & A.D.Wood. (2003). Stokes’ phenomenon and the absolutely continuous spectrum of one-dimensional Schrödinger operators. Journal of Computational and Applied Mathematics, vol. 171, no. 1–2, pg. 247-264. doi:10.1016/j.cam.2004.01.012