Non-real eigenvalues of singular indefinite Sturm-Liouville operators

Behrndt, Jussi; Katatbeh, Qutaibeh; Trunk, Carsten

doi:10.1090/S0002-9939-09-09964-X

Non-real eigenvalues of singular indefinite Sturm-Liouville operators
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by Jussi Behrndt, Qutaibeh Katatbeh and Carsten Trunk PDF

Proc. Amer. Math. Soc. 137 (2009), 3797-3806 Request permission

Abstract:

We study a Sturm-Liouville expression with indefinite weight of the form $\mathrm {sgn}(-d^2/dx^2+V)$ on $\mathbb {R}$ and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials $V$ with a certain behaviour at $\pm \infty$ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to $-d^2/dx^2+V$ in $L^2(\mathbb {R})$. The general results are illustrated with examples.

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