Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators

Abstract

We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter $Ω$ in a neighborhood of the real line. For real $Ω$, estimates are derived for all eigenvalue gaps uniformly in $Ω$. The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex $Ω$ is derived using the theory of slightly non-selfadjoint perturbations.