Eigenvalues of the Chandrasekhar–Page angular functions

Abstract

The Chandrasekhar–Page angular functions for the Dirac equation in the Kerr–Newman background are expanded as series of hypergeometric polynomials, and a three-term recurrence relation is derived for the coefficients in these series. This leads to a transcendental equation for the determination of the separation constant which is obtained initially as a power series and is then iterated by the method of Blanch and Bouwkamp.

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

“…Quantities on the upper rows which are not within our guaranteed error bounds are shaded. Table 2 contains computations of |λ −1 (± In this next experiment, we validate the numbers reported in reference [8] by means of sharpened eigenvalue enclosures determined from (3.2). This requires knowing beforehand some rough information about the position of the eigenvalues and the neighbouring spectrum.…”

“…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.…”

“…Various attempts at computing its spectrum have been considered in the past. 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].…”

“…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.…”