SCHRÖDINGER OPERATORS WITH δ AND δ ′-INTERACTIONS ON LIPSCHITZ SURFACES AND CHROMATIC NUMBERS OF ASSOCIATED PARTITIONS

Abstract. We investigate Schrödinger operators with δ and δ ′-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 Schrödinger operators with δ and δ ′-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 Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ ′-interactions. 1.

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