The spectrum of perturbations inside the Schwarzschild black hole

We study the spectrum of the bound state perturbations in the interior of the Schwarzschild black hole for the scalar, electromagnetic and gravitational perturbations. Demanding that the perturbations to be regular at the center of the black hole determines the spectrum of the bound state solutions. We show that our analytic expression for the spectrum is in very good agreement with the imaginary parts of the high overtone quasi normal mode excitations obtained for the exterior region. We also present a simple scheme to calculate the spectrum numerically to good accuracies.