Solutions of Second Order Schrödinger Wave Equations Near Static Black Holes and Strong Singularities of the Potentials
Igor M. Oliynyk
TL;DR
This work develops a geometric-analytic framework for Schrödinger operators with strongly singular potentials on noncompact (possibly incomplete) manifolds by introducing Range Control Neighborhoods (RCN) defined via an inner time metric. It establishes local nonnegativity of $H$ on RCNs, derives domain-of-dependence and energy inequalities for the Schrödinger wave equation, and proves existence, uniqueness, and global propagation results; these are then applied to Schwarzschild, Reissner–Nordström, and de Sitter spacetimes, showing horizons lie within RCNs of infinity and dynamics remain confined within RCNs. An essential self-adjointness analysis is obtained through a hyperbolic-equation approach, leveraging an admissible RCN cover and IMS localization. The Addendum extends the theory to strongly singular potentials of the form $-\beta^{2}/|x|^{2}$, giving sharp self-adjointness thresholds and connecting spectral properties to the geometry of the inner time metric.
Abstract
We consider a linear Schrödinger operator $H = -Δ+ V$ 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-Nordström, and de Sitter metrics - we study the wave equations for the corresponding Schrödinger 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.