On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

Abstract

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs), S mn (c, η), and their eigenvalues, λ mn , for arbitrary complex size parameter c in the asymptotic regime of large |c|, m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary arg(c), the PSWFs and their eigenvalues are accurately expressed by established prolate-type or oblate-type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points, c mn •;r , in the complex c-plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m, n, and arg(c). The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c, are derived for both the prolateand oblate-type asymptotic expansions and for both (n − m) even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of c mn •;r . We document this ordering for the specific case of arg(c) = π/4, which occurs for the diffusion equation in spheroidal coordinates. Some representative values of λ mn and S mn (c, η) for large, complex c are also given.

Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions

“…In our opinion, this provides some circumstantial evidence that the bending is not due to a branch cut. We refer to [7,10] for an extensive discussion of branch cuts and their effect on the prolate/oblate nature of the eigenvalues when s = 0.…”

Erratum: Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions [Phys. Rev. D73, 024013 (2006)]

“…In our opinion, this provides some circumstantial evidence that the bending is not due to a branch cut. We refer to [7,10] for an extensive discussion of branch cuts and their effect on the prolate/oblate nature of the eigenvalues when s = 0. The modulus of prolate eigenfunctions with increasing values of |c I | and s = −2 is plotted in Fig.…”

High-order late-time tail in a Kerr spacetime

“…The gravitational case is of particular interest given the recent detections of gravitational waves from black hole inspirals by the Laser Interferometer Gravitational-Wave Observatory [7,39], in which the ringdown stage was observed. It has been shown (see [30] and references therein) that the eigenvalues of the spheroidal wave equation possess square root branch cuts, typically chosen to be emanating radially outwards from points where two eigenvalues coalesce, and the eigenvalue problem is found to have a double root. Along these 'angular branch cuts', the two spheroidal eigenvalues are found to be analytic continuations of each other.…”