On mass-dependent spheroidal harmonics of spin one-half

Abstract

The angular functions in Chandrasekhar’s separation of the variables of Dirac’s equation in Kerr geometry are solved by an expansion procedure based on spin-weighted spherical harmonics. The characteristic values are obtained as a series in a σ, where a is the Kerr parameter and σ is the frequency. Closed expressions are obtained for the successive terms in the expansion. Numerical tables are provided.

Heun equation, Teukolsky equation, and type-D metrics

“…In the 70's, and 80's we find a large number of publications regarding the angular equation Chakrabarti (1984), and Seidel (1989). According to these references we will also name the solution of the angular equation as spin-weighted spheroidal function (SWSF).…”

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

“…For am = ±aω, the two canonical references on numerical approximations of λ n (κ; am, aω) are [8,10]. Suffern et al [8] derived an asymptotic expansion of the form…”

“…On the other hand, Chakrabarti [10] In both [8,10], the numerical estimation of λ n is achieved by means of a series expansions in terms of certain expressions involving aω and am, so it is to be expected that the approximations in both cases become less accurate as |aω| and |am| increase. However, no explicit error bounds are given in these papers and they seem to be quite difficult to derive.…”

“…Notably, a series expansion for λ in terms of a(m + ω) and a(m − ω) was derived in reference [8] by means of techniques involving continued fractions, see also [9]. A further asymptotic expansion in terms of aω and m/ω was reported in reference [10]. In both cases, however, no precise indication of the orders of magnitude of a reminder term was given.…”

“…Various numerical tests can be found in §5. We begin that section by describing details of the previous calculations reported in references [8,10]. These tests address the following: validity of the numerical values from references [8,10], sharpening of the eigenvalue bounds in the context of the quadratic method and optimal order of convergence.…”