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 obtained using the theory of slightly non-selfadjoint perturbations.

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

“…At this point, we make use of the fact that the scalar wave equation in the Kerr geometry is separable into ordinary differential equations for the radial and angular parts [4]. For the angular equation, we rely on the results of [10], where a spectral representation is obtained for the angular operator, and estimates for the eigenvalues and spectral projectors are derived. For the radial equation, we here derive rigorous estimates which are based on the semi-classical WKB approximation.…”

Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

“…In this section we shall illustrate our error estimates in examples for a real potential V and explain how the previous results from [3,2] fit into our framework. We first mention an additional structural relation which appears only for a real potential.…”

“…We consider the potential (inspired by the spheroidal wave operator [3]) ] (The precise choice of the boundary points is arbitrary and has no major effect. Moreover, since the factor 0.05 is so small, we simply adopted the terminology from quantum mechanics, disregarding the effect of the imaginary part of the potential.)…”

“…These equations can both be transformed into the Schrödinger form, which are coupled through two parameters ω and λ. For estimating the global behavior of the solutions uniformly in ω and λ, we were led to the idea of using invariant disk estimates [3,2]. When attempting to extend our analysis to gravitational waves in the framework of the Teukolsky equation [10], one of the difficulties that arises is that the potentials in the separated equations turn out to be complex.…”

“…The aim of the present paper is to introduce the method from a general point of view, and to illustrate it in typical examples. For an application to eigenvalue and gap estimates as well as to problems involving parameters we refer to [3] and [2].…”