The Stokes constants for a cluster of transition points

Abstract

AbstractThe connection problems associated with the one-dimensional Schrödinger equation in the presence of a general isolated cluster containing an unspecified number of complex transition points in unspecified positions can be studied by means of the phase-integral method developed by Fröman and Fröman. Any anti-Stokes line, i.e. any line in the complex z-plane on which the solutions behave as travelling waves with constant flow, must asymptotically (i.e. in the limit of large values of |z|) point in one of m +2 possible directions, which divide the region around the cluster into m +2 sectors, where m is the degree of the cluster. The tracing of these waves from an anti-Stokes line, bounding a sector, to an anti-Stokes line constituting the other boundary of the same sector is expressed by means of the Stokes constant for the sector in question. This paper examines the relation between these m + 2 Stokes constants in the general case when the transition points in the cluster may also be close-lying in the sense that it is impossible to treat them individually, when the solutions are traced. Under the assumption that the effective potential in the Schrodinger equation is a regular analytic function in a sufficiently large region containing the cluster, it is shown that the m + 2 Stokes constants are in general constrained by three algebraic relations, which are obtained for arbitrary m. The cases m = 1, 2, 3 and 4 are worked out in detail.