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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectral bounds for singular indefinite Sturm-Liouville operators with $ L^1$-potentials


Authors: Jussi Behrndt, Philipp Schmitz and Carsten Trunk
Journal: Proc. Amer. Math. Soc. 146 (2018), 3935-3942
MSC (2010): Primary 34L15, 47E05
DOI: https://doi.org/10.1090/proc/14059
Published electronically: April 18, 2018
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Abstract: The spectrum of the singular indefinite Sturm-Liouville operator

$\displaystyle A=\operatorname {sgn}(\cdot )\bigl (-\tfrac {d^2}{dx^2}+q\bigr )$    

with a real potential $ q\in L^1(\mathbb{R})$ covers the whole real line, and, in addition, non-real eigenvalues may appear if the potential $ q$ assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction have been obtained. In this paper the bound

$\displaystyle \vert \lambda \vert \leq \Vert q\Vert _{L^1}^2$    

on the absolute values of the non-real eigenvalues $ \lambda $ of $ A$ is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the $ L^1$-norm of the negative part of $ q$.

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Additional Information

Jussi Behrndt
Affiliation: Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria
Email: behrndt@tugraz.at

Philipp Schmitz
Affiliation: Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
Email: philipp.schmitz@tu-ilmenau.de

Carsten Trunk
Affiliation: Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany – and – Instituto Argentino de Matemática “Alberto P. Calderón” (CONICET), Saavedra 15, (1083) Buenos Aires, Argentina
Email: carsten.trunk@tu-ilmenau.de

DOI: https://doi.org/10.1090/proc/14059
Keywords: Non-real eigenvalue, indefinite Sturm-Liouville operator, Krein space, Birman-Schwinger principle
Received by editor(s): September 14, 2017
Received by editor(s) in revised form: December 7, 2017
Published electronically: April 18, 2018
Communicated by: Wenxian Shen
Article copyright: © Copyright 2018 American Mathematical Society