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A Poisson Formula for the Wave Propagator on Schwarzschild-de Sitter Backgrounds

Izak Oltman, Ben Pineau

TL;DR

This work extends Poisson-type trace formulas for wave propagators to Schwarzschild–de Sitter backgrounds by treating a class of exponentially decaying, non-compactly supported potentials. It develops a detailed 1D scattering framework, proving meromorphic continuation of the resolvent and constructing incoming/outgoing solutions, then applies the Birman–Krein trace formula to relate wave traces to the resonance spectrum. The approach separates the scattering data into gamma-term contributions and a function $F(\lambda)$ whose zeros encode resonances, and introduces renormalization terms $A_\pm$ to account for non-compact tails. The result is a global trace formula for the SdS metric obtained by summing over angular momentum and interpreting the renormalized trace as a well-defined distribution tied to the quasinormal mode spectrum.

Abstract

This paper proposes a Poisson formula for the wave propagator of the Schwarzschild--de Sitter (SdS) metric. That is done by proving a Poisson formula relating wave propagators and scattering resonances for a class of non-compactly supported potentials on the real line. That class includes the Regge--Wheeler potentials obtained from separation of variables for SdS. The novelty lies in allowing non-compact supports -- all exact Poisson formulae of Lax--Phillips, Melrose, and other authors required compactness of the support of the perturbation.

A Poisson Formula for the Wave Propagator on Schwarzschild-de Sitter Backgrounds

TL;DR

This work extends Poisson-type trace formulas for wave propagators to Schwarzschild–de Sitter backgrounds by treating a class of exponentially decaying, non-compactly supported potentials. It develops a detailed 1D scattering framework, proving meromorphic continuation of the resolvent and constructing incoming/outgoing solutions, then applies the Birman–Krein trace formula to relate wave traces to the resonance spectrum. The approach separates the scattering data into gamma-term contributions and a function whose zeros encode resonances, and introduces renormalization terms to account for non-compact tails. The result is a global trace formula for the SdS metric obtained by summing over angular momentum and interpreting the renormalized trace as a well-defined distribution tied to the quasinormal mode spectrum.

Abstract

This paper proposes a Poisson formula for the wave propagator of the Schwarzschild--de Sitter (SdS) metric. That is done by proving a Poisson formula relating wave propagators and scattering resonances for a class of non-compactly supported potentials on the real line. That class includes the Regge--Wheeler potentials obtained from separation of variables for SdS. The novelty lies in allowing non-compact supports -- all exact Poisson formulae of Lax--Phillips, Melrose, and other authors required compactness of the support of the perturbation.

Paper Structure

This paper contains 11 sections, 20 theorems, 219 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Trace formula for a class non-compactly supported potentials.
  3. Regge--Wheeler potentials
  4. Outline of proof
  5. Analysis of incoming and outgoing solutions
  6. Incoming and outgoing solutions
  7. Meromorphic continuation of the resolvent
  8. The Scattering Matrix
  9. Applying the Birman--Kreı̆n trace formula
  10. Proof of Lemma 3.3
  11. Birman-Kreı̆n trace formula for exponentially decaying potentials

Key Result

Proposition 1.1

The resolvent of $P_V \coloneq D_x^2 + V ( x )$, $D_x \coloneq \partial_x /i ,$ continues meromorphically to

Figures (1)

  • Figure 1: Comparison between a numerical approximation of the flat trace for the Pöschl–Teller potential (with $\ell = 1$) and the analytic expression in \ref{['eq:243']}. The numerical trace is computed in MATLAB by truncating and discretizing the spatial domain, evolving a normalized Gaussian centered at each grid point, $x_i$, under the cosine propagator $\cos(t\sqrt{P_{V_{\mathrm{PT}}}})$, subtracting the free wave evolution of the same Gaussian, and evaluating at $x_i$. This approximates $U_{V_{\rm{PT}}}(t,x_i,x_i)-U_{0}(t,x_i,x_i)$ (the Schwartz kernels). Computing the sum of these terms, multiplied by $x_{i} - x_{i-1}$ will approximate the flat trace.

Theorems & Definitions (41)

  • Proposition 1.1
  • Theorem : Main result
  • Example 1.1: Pöschl-Teller potential
  • Theorem : Trace formula for each spherical harmonic
  • Remark 1.1
  • Theorem 1: Global trace formula
  • Definition 2.1: Outgoing solution
  • Definition 2.2: Incoming solution
  • Proposition 2.3
  • proof
  • ...and 31 more