Schrödinger operators with $\delta$-interactions supported on conical surfaces

Abstract

We investigate the spectral properties of self-adjoint Schrödinger operators with attractive $\delta$-interactions of constant strength $\alpha >0$ supported on conical surfaces in ${\mathbb{R}}^3$. It is shown that the essential spectrum is given by $[-{\alpha }^2/4,+\infty )$ and that the discrete spectrum is infinite and accumulates to $-{\alpha }^2/4$. Furthermore, an asymptotic estimate of these eigenvalues is obtained.

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Cited by in corpus (18)

• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces
• Generalized interactions supported on hypersurfaces
• On the negative spectrum of the Robin Laplacian in corner domains
• Dirichlet spectrum of the Fichera layer
• On the discrete spectrum of Robin Laplacians in conical domains
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• Discrete spectrum of interactions concentrated near conical surfaces
• A spectral isoperimetric inequality for cones
• Spectral asymptotics of the Dirichlet Laplacian in a conical layer
• Eigenvalue counting function for Robin Laplacians on conical domains
• Spectral properties of soft quantum waveguides
• A general review on the NLS equation with point-concentrated nonlinearity
• Magnetic Laplacian in sharp three dimensional cones
• Attractive conical surfaces create infinitely many bound states
• Spectral asymptotics for $\delta$-interactions on sharp cones
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• 2D Schrödinger operators with singular potentials concentrated near curves