  In addition to being the eigenfunctions of the restricted Fourier operator,
the angular spheroidal wave functions of the first kind of order zero and
nonnegative integer characteristic exponents are the solutions of a singular
self-adjoint Sturm-Liouville problem. The running time of the standard
algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues
grows with both bandlimit and characteristic exponent. Here, we describe a new
approach whose running time is bounded independent of these parameters.
Although the Sturm-Liouville eigenvalues are of little interest themselves, our
algorithm is a component of a fast scheme for the numerical evaluation of the
prolate spheroidal wave functions developed by one of the authors. We
illustrate the performance of our method with numerical experiments.
