Bernstein spectral method for quasinormal modes and other eigenvalue problems

TL;DR

This paper introduces Bernstein polynomials as a useful basis for spectral methods in eigenvalue problems, presents a new software package, and demonstrates its effectiveness in quantum mechanics and black hole quasinormal mode computations.

Contribution

The work highlights the advantages of Bernstein polynomials for boundary condition handling and provides a new user-friendly spectral solver package, SpectralBP.

Findings

Successfully applied to quantum mechanics models

Accurately computed scalar and gravitational quasinormal modes

Achieved excellent agreement with known results

Abstract

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 \texttt{SpectralBP}, that implements Berstein-polynomial-based pseudospectral 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 purely imaginary modes such as the algebraically special modes for Schwarzschild gravitational perturbations.