Microlocal Spectral Bifurcation in Schwarzschild–de Sitter Geometry

We develop a complete semiclassical resolvent theory for the Regge-Wheeler operator on Schwarzschild-de Sitter (SdS) spacetimes. We demonstrate that in the strictly physical regime, the photon sphere is permanently shielded from geometric degeneracy, maintaining strict normal hyperbolicity. By analytically continuing the parameter space beyond the Nariai boundary, we identify an exact algebraic locus P_c where the spectral gap of the linearized Hamiltonian flow completely collapses. Via microlocal reduction to a k=3 (cusp) normal form with outgoing boundary conditions, we establish a non-diagonalizable Jordan block of size 3. An asymptotic WKB suppression mechanism yields the two-sided resolvent estimate ||R_h(P)|| ~ h-6/5 |P-P_c|-1/2, valid in the semiclassical limit h->0. In the double scaling regime |P-P_c| ~ h4/5, the resolvent remains uniformly bounded by O(h-6/5) and the resonance spacing satisfies Delta-omega ~ h6/5. The critical frequency omega_c is purely imaginary; the three coalescing resonances lie on the imaginary axis at P_c, and the splitting occurs at angle -pi/10 from the imaginary axis.