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.

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• Lower bounds on the lowest spectral gap of singular potential Hamiltonians
• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction on a planar loop

Cited by in corpus (17)

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• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces
• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Generalized interactions supported on hypersurfaces
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• A spectral isoperimetric inequality for cones
• On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries
• On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• Optimization of the lowest eigenvalue of a soft quantum ring
• Gap control by singular Schrödinger operators in a periodically structured metamaterial
• Scattering from local deformations of a semitransparent plane
• Smilansky-Solomyak model with a ${\delta }^{\prime }$-interaction
• Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions
• 2D Schrödinger operators with singular potentials concentrated near curves
• On Schrödinger operators with ${\delta }^{\prime }$-potentials supported on star graphs
• Trace Hardy inequality for the Euclidean space with a cut and its applications