A Spectral Representation for Spin-Weighted Spheroidal Wave Operators with Complex Aspherical Parameter

TL;DR

This paper develops a spectral decomposition for spin-weighted spheroidal wave operators with complex parameters, addressing non-symmetry and complex spectra, and establishing bounds and completeness of the spectral representation.

Contribution

It introduces a spectral decomposition framework for non-symmetric operators with complex spectra, including bounds on Jordan chains and proof of completeness.

Findings

Spectral decomposition constructed for complex aspherical parameters

Uniform bounds established for Jordan chain lengths

Completeness of the spectral decomposition proven

Abstract

A family of spectral decompositions of the spin-weighted spheroidal wave operator is constructed for complex aspherical parameters with bounded imaginary part. As the operator is not symmetric, its spectrum is complex and Jordan chains may appear. We prove uniform upper bounds for the length of the Jordan chains and the norms of the idempotent operators mapping onto the invariant subspaces. The completeness of the spectral decomposition is proven.