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Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators

Finster, Felix and Schmid, Harald (2006) Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators. Journal für die reine und angewandte Mathematik 601, pp. 71-107.

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Other URL: http://arxiv.org/PS_cache/math-ph/pdf/0405/0405010v4.pdf

Abstract

The spheroidal wave operator $\cal A$ is a linear elliptic operator of second order with smooth coefficients on the unit sphere $S^2$ . Using angular variables $\vartheta \in (0,\pi)$ and $\phi\in [0,2\pi)$ this operator may be written in the form ${\cal A}=-{d\over{d \cos \vartheta}}\sin^2\vartheta {d\over{d \cos \vartheta}} + {1\over {\sin^2 \vartheta}}\left(\Omega\sin^2\vartheta+k\right)^2.$ Here $\Omega\in{\Bbb C}$ is the aspherical parameter. The authors consider the operator $\cal A$ in the Hilbert space ${\cal H}=L^2\left((0,\pi),\sin \vartheta d\vartheta\right)$ with boundary conditions $\lim_{\vartheta\to 0,\pi}\Theta^\prime(\vartheta)=0 \text{\ if \ } k=0, \qquad \lim_{\vartheta\to 0,\pi}\Theta(\vartheta)=0 \text{\ if \ } k\neq 0.$ It is proved that the spectral representation for $\cal A$ 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.