Semi-classical Imprint of Horizon Induced Instability

TL;DR

This paper analyzes the spectral properties of an inverted harmonic oscillator on a circle, linking classical instability to quantum thermalization through semi-classical methods.

Contribution

It provides a rigorous spectral analysis connecting classical Lyapunov instability with quantum thermalization, addressing gaps in previous heuristic approaches.

Findings

Computed density of states using stationary phase approximation.

Linked classical instability to quantum thermalization.

Closed conceptual gaps in previous literature.

Abstract

We consider an inverted harmonic oscillator in the space $L^{2} (\mathbb{S})$ of square-integrable functions on the circle $\mathbb{S}$ and compute its density of states employing the stationary phase approximation. Our computation is based on an oscillatory integral representation of the Schwartz kernel of the time-evolution operator. This demonstrates thermalisation as a semi-classical manifestation of the classical Lyapunov instability -- reported earlier in [Phys. Rev. D 102, 044006; Phys. Rev. D 102, 124047] using heuristic analytic continuation. Our spectral analysis of the Hamiltonian points out and closes the conceptual and mathematical gaps in the preceding literature.