Eigenvalue problem for a set of coupled Schrödinger like ODEs

The numerical solution of an eigenvalue problem for a set of ODEs may be non-trivial when high accuracy is needed and the interval of the independent variable extends to infinity. In that case, efficient asymptotics are needed at infinity to produce the initial conditions for numerical integration. Here such asymptotics are found for a set of N coupled 1D Schrödinger like ODEs in r, 0 ≤ r < ∞. This is a generalization of the well known phase integral approximation used for N = 1. Calculations are performed for N = 2; the ODEs describe small vibrations of a single quantum vortex in a Bose–Einstein condensate, where a critical situation arises in the long-wavelength limit, k → 0. The calculations were aimed at clarifying certain discrepancies in theoretical results pertaining to this limit. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)