Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

Jussi Behrndt, Pavel Exner, Vladimir Lotoreichik

Type: Review

Publication Date: 2014-07-31

Citations: 64

DOI: https://doi.org/10.1142/s0129055x14500159

Abstract

We investigate Schr\"odinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schr\"odinger operators with \delta- and \delta'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schr\"odinger operators and, in particular, it allows to transform known results for Schr\"odinger operators with \delta-interactions to Schr\"odinger operators with \delta'-interactions.

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+	Spectral Theory of Self-Adjoint Operators in Hilbert Space	1987	M. Sh. Birman M. Z. Solomjak
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+	Some properties of the Hamiltonian describing a finite number of δ’-interactions with support on concentric spheres	1988	Juma Shabani
+	Schrödinger-Operators with Singular Interactions	1994	Johannes F. Brasche Pavel Exner Yu. A. Kuperin P. Šeba
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