Effective Angular Asymptotics and the Sharp $D^{-3}$ Horoconvex Gap Scale

Abstract

We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \[V\] on \[\mathbb S^{n-1}\]. For graph domains \[r<R-V(\theta)\], the first band satisfies \[ \lambda_{j+1}=\alpha^2+\frac{\pi^2}{R^2} +\frac{2\pi^2}{R^3}\bigl(\eta_j(T_n+V)-b_0\bigr)+o(R^{-3}), \qquad j=0,1, \] where \[T_n\] is the nonlocal spherical operator with multiplier \[\psi(\ell+\alpha)-\psi(\alpha)\]. Consequently the horoconvex fundamental gap has the sharp \[D^{-3}\] polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula \[ 16\pi^2\inf_{V\in\mathcal A_n} \bigl(\eta_1(T_n+V)-\eta_0(T_n+V)\bigr). \] Geodesic balls realize the polynomial scale, but an explicit admissible axial perturbation lowers the reduced leading-constant value at first order.