Methods and Applications of Analysis

Volume 23 (2016)

Number 1

A spectral representation for spin-weighted spheroidal wave operators with complex aspherical parameter

Pages: 35 – 118



Felix Finster (Fakultät für Mathematik, Universität Regensburg, Germany)

Joel Smoller (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


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.


non-selfadjoint spectral problem, Sturm–Liouville equation, spin-weighted spheroidal wave operator, angular Teukolsky operator

2010 Mathematics Subject Classification

34B05, 34L10, 34L15, 47E05

Published 9 March 2016