Abstract

Stokes phenomenon and the absolutely continuous spectrum of one-dimensional Schroedinger operators D.J.Gilbert and A.D.Wood It is well known that the Airy functions $Ai(-x - \lambda)$ and $Bi(-x - \lambda)$ form a fundamental set of solutions for the differential equation begin{displaymath} L u(x) := - u''(x) - x u(x) = \lambda u(x), \hspace{1cm} 0 \leq x < \infty, \hspace{1cm} \lambda \in \Re, end{displaymath} and that the spectrum of the associated selfadjoint operator is purely absolutely continuous for any choice of boundary condition at $x = 0$. Also widely known is the fact that the semi-axis $[\lambda, \infty)$ is a Stokes line of the differential equation $L u = \lambda u$ for each fixed value of the spectral parameter $\lambda$. In this paper, we show that the connection between the existence of Stokes lines on the real axis and points of the absolutely continuous spectrum holds under much more general circumstances. Further correlations, connecting the Stokes phenomenon with the asymptotic behaviour of solutions of $L u = \lambda u$ at infinity and with the boundary behaviour of the Titchmarsh-Weyl $m$-function at $\lambda$, are also deduced.

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