Generalized interactions supported on hypersurfaces

Abstract

We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface $\Sigma \subset {\mathbb{R}}^n,n\ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of $\Sigma$ the interaction is characterized by four real parameters, the earlier studied case of $\delta$- and ${\delta }^{\prime }$-interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

References in corpus (14)

• Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials
• Bound States in Curved Quantum Layers
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Mean curvature bounds and eigenvalues of Robin Laplacians
• Asymptotic eigenvalue estimates for a Robin problem with a large parameter
• Bound states due to a strong $\delta$ interaction supported by a curved surface
• Some properties of the one-dimensional generalized point interactions (a torso)
• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Spectra of soft ring graphs
• Tunneling between corners for Robin Laplacians
• Resonance asymptotics in the generalized Winter model
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• On geometric perturbations of critical Schrödinger operators with a surface interaction

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• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions
• 2D Schrödinger operators with singular potentials concentrated near curves
• Scattering properties of two singularly interacting particles on the half-line
• On Schrödinger operators with ${\delta }^{\prime }$-potentials supported on star graphs
• Trace Hardy inequality for the Euclidean space with a cut and its applications
• A non-degeneracy theorem for interacting fermions in one dimension