East Carolina University
Greenville, North Carolina, U. S. A.
The wave equation $\nabla^\alpha \nabla_\alpha \psi \ = \ 0 $ is studied in the exterior of a Schwarzschild black hole, $r>2M$, defined by the metric $ds^2 = \mu dt^2 - \mu^{-1} dr^2 - r^2 d\Omega $, $\mu \equiv (1 - 2M/r)$. By assuming stationary and spherically-symmetric solutions $\psi = e^{i \omega t} r^{-1} \phi (r) Y_\ell (\Omega )$, the wave equation reduces to the Schr\"odinger equation $(-\partial_{r_*}^2 + V(r,\ell ))\phi = \omega^2 \phi $, where $r_* = \int \mu^{-1} dr = r + 2M \ln (r/2M - 1)$. The potential $V$ has a single maximum near $r=3M$ for each $\ell \in {\bf N}$, so a family of top resonances is expected to exist. It is demonstrated that there are spectral resonances $z_\ell^k \sim V_{max}^\ell - i\ell (2k-1)/9M^{2} $ where $k\in {\bf N}$ is a parameter of the harmonic oscillator, and where $V_{max}^\ell \equiv V(r_{max}^\ell , \ell ) \sim {{\ell (\ell + 1)}\over {27M^2}} \ + \ {2 \over {81 M^2}} $. The resonance state orbits near the radius $r_{max}^\ell \sim 3M (1 - (3 \ell )^{-2} )$ for large $\ell >0$. A modification of the standard complex scaling technique is required for the analysis.