Semi-classical Imprint of Horizon Induced Instability

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.