Distribution of resonances for spherical black holes

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

“…The results are described in Appendix B. One should note that the quantization condition of [41] was stated up to O(l −1 ) error, while Theorem 1 has error O(l −∞ ); we demonstrate numerically that increasing the order of the quantization condition leads to a substantially better approximation.…”

“…To reduce the bottom of the well problem to the barrier-top problem, we formally rescale in the complex plane, introducing the parameter y = e iπ/4 y, so that (y ) 2 = iy 2 . We do not provide a rigorous justification for such an operation; we only note that the WKB solution of (B.6) looks like e ic(y ) 2 a = e −cy 2 a near y = 0 for some positive constant c; therefore, it is exponentially decaying away from the origin, reminding one of the exponentially decaying Gaussians featured in the bottom of the well asymptotics (see for example [21,Section 3] or the discussion following [41,Proposition 4.3]). There is a similar calculation of the bottom of the well resonances based on quantum Birkhoff normal form; see for example [15].…”

“…(Note that we take a different path here than [41] and [10], using only real microlocal analysis near the trapped set, which immediately gives polynomial resolvent bounds).…”

“…Another difference between (0.2) and the quantization condition of [41] is the extra parameter k, resulting from the lack of spherical symmetry of the problem. In fact, for a = 0 each pole in (0.3) has multiplicity 2l + 1; for a = 0 this pole splits into 2l + 1 distinct QNMs, each corresponding to its own value of k, the angular momentum with respect to the axis of rotation.…”

“…QNMs of Schwarzschild-de Sitter metric were then investigated by Sá Barreto-Zworski [41], resulting in the lattice of pseudopoles given by (0.3) below. For this case, Bony-Häfner [10] established polynomial cutoff resolvent estimates and a resonance expansion, Melrose-Sá Barreto-Vasy [38] obtained exponential decay for solutions to the wave equation up to the event horizons, and Dafermos-Rodnianski [17] used physical space methods to obtain decay of linear waves better than any power of t.…”