Solutions of Second Order Schr\"odinger Wave Equations Near Static Black Holes and Strong Singularities of the Potentials

TL;DR

This paper investigates Schrödinger operators with strongly singular potentials near black hole horizons, establishing conditions for semiboundedness, and analyzing wave solutions' behavior in these extreme geometries.

Contribution

It introduces new geometric conditions called range control neighborhoods (RCN) and studies wave equations near black hole horizons using an inner time metric.

Findings

Black hole horizons belong to RCNs of infinity.

Solutions of wave equations remain in RCNs indefinitely.

Semiboundedness of Schrödinger operators is established under new conditions.

Abstract

We consider a linear Schr\"odinger operator $H = -\Delta + V$ with a strongly singular potential $V$ not bounded from below on a non-compact incomplete Riemannian manifold $M$. We assume that the negative part of potential $V_{-}$ is measurable, and it does not necessarily belong to either local Kato or Stummel classes, and we define new geometric conditions on the growth of $V_{-}$ in a special $\textit{range control neighborhood (RCN)}$ such that $H$ is semibounded from below on functions compactly supported in these neighborhoods. We define RCN by means of $\textit{an inner time metric}$ which estimates the minimal time for a classical particle to travel between any two points on $M$, and we assume that $M$ is complete w.r.t. this metric, i.e. the potential $V$ is classically complete on $M$. For the corresponding Cauchy problem of the wave equation $u_{tt} + Hu = 0$, we define locally a Lorentzian metric such that its light cone is formed along the minimizing curves with respect to the inner time metric, where both an energy inequality and uniqueness of solutions hold. Inversely, for well-known Lorentzian metrics of static black holes - Schwarzschild, Reissner-Nordstr\"om, and de Sitter metrics - we study the wave equations for the corresponding Schr\"odinger operators, and we show that the event horizons of these black holes belong to the RCNs of infinity w.r.t. the inner time metrics, and that all solutions of the mixed problems stay in these neighborhoods indefinitely long.