Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of black hole quasinormal modes. Most of these spectral codes are based on standard Chebyshev, Fourier, or some other orthogonal basis functions. In this work we highlight the usefulness of a relatively unknown set of non-orthogonal basis functions, known as Bernstein polynomials, and their advantages for handling boundary conditions in ordinary differential eigenvalue problems. We also report on a new user-friendly package, called SpectralBP, that implements Berstein-polynomial-based spectral routines for eigenvalue problems. We demonstrate the functionalities of the package by applying it to a number of model problems in quantum mechanics and to the problem of computing scalar and gravitational quasinormal modes in a Schwarzschild background. We validate our code against some known results and achieve excellent agreement. Compared to continued-fraction or series methods, global approximation methods are particularly well-suited for computing the algebraically special modes for gravitational perturbations of the Schwarzschild geometry. We demonstrate this by reporting the most accurate numerical calculation of these modes to date, achieved with only modest resources.

We revisit the peculiar electromagnetic quasinormal mode spectrum of an asymptotically anti-de Sitter Schwarzschild black hole. Recent numerical calculations have shown that this quasinormal mode spectrum becomes purely overdamped at some critical black hole sizes, where the spectrum also bifurcates. In this paper, we shed light on unnoticed and unexplained properties of this spectrum, by exploiting some novel analytic results for the large black hole limit. In particular, we demonstrate that the quasinormal mode spectrum is self-similar in this limit. We take advantage of this self-similarity to derive a precise analytic expression for the locations of the bifurcations, in which a surprising Feigenbaum-like constant appears. Also in this large black hole limit, we show that the differential operator is of Sturm-Liouville type and thus self-adjoint. We derive an exact solution for its spectrum and eigenfunctions, and find that large black holes cannot be made to vibrate with electromagnetic perturbations, independently of the boundary conditions imposed at spatial infinity. Finally, we characterize the insensitivity of the spectrum to different boundary conditions by analyzing the expansion of the quasinormal mode spectrum around the large black hole limit.

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