SEMICLASSICAL MICROLOCAL ANALYSIS OF RESONANT STATES AND THE FRACTAL WEYL LAW FOR NON-SELF-ADJOINT OPERATORS IN REGIMES OF ERGOSPHERIC TRAPPING

Mathematical Applications of Science Fiction We present a rigorous semiclassical analysis of the distribution of scattering resonances for non-self-adjoint operators governing wave propagation on Lorentzian manifolds with an ergosphere. Specifically, we study the spectral behavior of a semiclassical differential operator in the limit as the semiclassical parameter approaches zero, where the trapping set exhibits hyperbolic dynamics with a fractal structure. By constructing a refined escape function adapted to the ergospheric geometry and applying the Fourier–Bros–Iagolnitzer (FBI) transform, we derive a Fractal Weyl Law describing the density of resonances within the semiclassical regime. We show that the number of resonances near the real axis scales with a power determined by the Hausdorff dimension of the trapped set. Furthermore, we establish ten fundamental lemmas describing the propagation of singularities through the ergoregion, thereby providing a rigorous bridge between classical general relativity and non-Hermitian spectral theory.