Stokes phenomenon for the prolate spheroidal wave equation

The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2:(t2−τ2)φ″+2tφ′+σ2t2φ=μφ.(t2−τ2)φ″+2tφ′+σ2t2φ=μφ.In this article we study the formal solutions of this equation in the neighborhood of the singularities (the regular ones ±τ, and the irregular one, at infinity) and perform some numerical experiments on the computation of Stokes matrices and monodromy, using formal/numerical algorithms we developed recently in the Maple package Desir. This leads to the following conjecture: the series appearing in the formal solutions at infinity, depending on the parameter μ, are in general divergent; they become convergent for some particular values of the parameter, corresponding exactly to the eigenvalues of the prolate operator.We give the proof of this result and its interpretation in terms of differential Galois groups.