Grid method for computation of generalized spheroidal wave functions based on discrete variable representation

We present an efficient and accurate grid method for computations of eigenvalues and eigenfunctions of the generalized spheroidal wave equation. Different from previous studies, our method is based on the expansion of the spheroidal wave function by discrete-variable-representation basis functions constructed from the associated Legendre polynomials. The differential operator can be expressed analytically on the grid points, which are the zeros of the associated Legendre polynomials. The resultant potential matrix is simply diagonal and evaluated directly on the same grid. The corresponding differential equation is thus converted to an eigenvalue problem of a small matrix, whose eigenvalues and eigenvectors are converged very fast. The wave functions can then be evaluated accurately at any desired point from the expansion formula with the computed eigenvectors. Compared to previous methods, our method is direct and efficient for any parameter c, either small or large.