Natural polynomials for Kerr quasinormal modes

We present a polynomial basis that exactly tridiagonalizes Teukolsky’s radial equation for quasinormal modes. These polynomials naturally emerge from the radial problem, and they are “canonical” in that they possess key features of classical polynomials. Our canonical polynomials may be constructed using various methods, the simplest of which is the Gram-Schmidt process. In contrast with other polynomial bases, our polynomials allow for Teukolsky’s radial equation to be represented as a simple matrix eigenvalue equation. We expect that our polynomials will be useful for better understanding the Kerr quasinormal modes’ properties, particularly their prospective spatial completeness and orthogonality. We show that our polynomials are closely related to the confluent Heun and Pollaczek-Jacobi type polynomials. Consequently, our construction of polynomials may be used to tridiagonalize other instances of the confluent Heun equation. We apply our polynomials to a series of simple examples, including: (1) the high accuracy numerical computation of radial eigenvalues, (2) the evaluation and validation of quasinormal mode solutions to Teukolsky’s radial equation, and (3) the use of Schwarzschild radial functions to represent those of Kerr. Along the way, a potentially new concept, “polynomial/nonpolynomial duality,” is encountered and applied to show that some quasinormal mode separation constants are well approximated by confluent Heun polynomial eigenvalues. We briefly discuss the implications of our results on various topics, including the prospective spatial completeness of Kerr quasinormal modes.