Mathematical Physics
[Submitted on 29 Jun 2013
]
Title: Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
Title: 薛定谔算子在 Lipschitz 曲面上的 δ 和 δ' 相互作用以及相关分划的色数
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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