Schr\"odinger operators with \delta- and \delta'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

TL;DR

This paper explores the relationship between the spectral properties of Schr"odinger operators with elta- and elta'-interactions on Lipschitz surfaces and the combinatorial chromatic number of the partitioned space, providing new spectral inequalities.

Contribution

It introduces an operator inequality based on optimal coloring that links the spectral properties of elta- and elta'-interactions, enabling transfer of results between these operators.

Findings

Established an operator inequality involving the chromatic number.

Linked spectral properties of elta- and elta'-interactions.

Enabled transfer of spectral results between different interaction types.

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.