Exact quasinormal residues and double poles from hypergeometric connection formulas

TL;DR

This paper introduces a unified algebraic method using hypergeometric connection formulas to analyze the pole structure and quasinormal modes in boundary value problems, enabling explicit calculations of residues and double poles.

Contribution

It develops an explicit quantization function framework that encodes boundary conditions and identifies double poles algebraically, advancing spectral analysis in hypergeometric problems.

Findings

Validated against BTZ black hole spectrum

Provides analytic criteria for double-pole quasinormal modes

Diagnoses nearly double-pole excitations in specific limits

Abstract

We develop a unified mathematical method for the pole structure of frequency-domain Green's functions and the associated quasinormal spectra in radial boundary value problems reducible to the Gauss hypergeometric equation. By systematically employing connection formulas for Kummer solutions, we construct an explicit quantization function that encodes arbitrary linear asymptotic boundary conditions. We demonstrate that the frequency-dependent spectral factor entering the residue formula is controlled algebraically by the closed-form Digamma derivative of this quantization function, bypassing integral evaluation. Furthermore, we establish the simultaneous vanishing of the quantization function and its first derivative as a direct algebraic criterion for double-pole QNMs. The formalism is successfully benchmarked against the exact BTZ black hole spectrum and provides an analytic diagnostic for the exceptional lines and nearly double-pole excitations in the Nariai/P\"oschl-Teller limit.