Beyond quasinormal modes: a complete mode decomposition of black hole perturbations

We show that retarded Green's functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or, Euclidean) mode sum converges. The two regions are separated by a lightcone which scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated to the early time region. We illustrate our results for P\"oschl-Teller, BTZ, and Schwarzschild. In the case of Schwarzschild, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $\Lambda\to 0^+$, and exploit recent exact solutions to the Heun connection problem. In each case we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.