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: ( t 2 − τ 2 ) φ ″ + 2 t φ ′ + σ 2 t 2 φ = μ φ . s 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. e the proof of this result and its interpretation in terms of differential Galois groups.