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In addition to being the eigenfunctions of the restricted Fourier operator,the angular spheroidal wave functions of the first kind of order zero andnonnegative integer characteristic exponents are the solutions of a singularself-adjoint Sturm-Liouville problem. The running time of the standardalgorithm for the numerical evaluation of their Sturm-Liouville eigenvaluesgrows with both bandlimit and characteristic exponent. Here, we describe a newapproach whose running time is bounded independent of these parameters.Although the Sturm-Liouville eigenvalues are of little interest themselves, ouralgorithm is a component of a fast scheme for the numerical evaluation of theprolate spheroidal wave functions developed by one of the authors. Weillustrate the performance of our method with numerical experiments.

An $\mathcal{O}\left(1\right)$ algorithm for the numerical evaluation of  the Sturm-Liouville eigenvalues of the spheroidal wave functions of order  zero