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Optimization of the lowest eigenvalue for leaky star graphs

Pavel Exner Vladimir Lotoreichik

Spectral Theory and Operator Algebra mathscidoc:1910.43228

Proceedings of the conference Mathematical Results in Quantum Physics(QMath13, Atlanta 2016, 187-196, 2016.10
We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrdinger operator with an attractive -interaction of a fixed strength, the support of which is a star graph with finitely many edges of an equal length L(0,]. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.
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@inproceedings{pavel2016optimization,
  title={Optimization of the lowest eigenvalue for leaky star graphs},
  author={Pavel Exner, and Vladimir Lotoreichik},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133947502762757},
  booktitle={Proceedings of the conference Mathematical Results in Quantum Physics(QMath13, Atlanta 2016},
  pages={187-196},
  year={2016},
}
Pavel Exner, and Vladimir Lotoreichik. Optimization of the lowest eigenvalue for leaky star graphs. 2016. In Proceedings of the conference Mathematical Results in Quantum Physics(QMath13, Atlanta 2016. pp.187-196. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133947502762757.
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