Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators

TL;DR

This paper develops a spectral representation for oblate spheroidal wave operators, providing uniform eigenvalue gap estimates for real parameters and extending the analysis to complex parameters using non-selfadjoint perturbation theory.

Contribution

It introduces a holomorphic spectral representation for the operator and establishes uniform eigenvalue gap estimates, advancing understanding of non-selfadjoint spheroidal operators.

Findings

Spectral representation is holomorphic in the aspherical parameter near the real line.

Eigenvalue gaps are estimated uniformly for all real aspherical parameters.

Spectral analysis extends to complex parameters using non-selfadjoint perturbation theory.

Abstract

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