Stokes’ phenomenon and the absolutely continuous spectrum of one-dimensional Schrödinger operators

摘要

It is well known that the Airy functions, Ai(−x−μ) and Bi(−x−μ), form a fundamental set of solutions for the differential equationLu(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.