Discrete spectrum of interactions concentrated near conical surfaces

Abstract

We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrödinger operators with attractive $\delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.

18 pages

References in corpus (3)

• Leaky Quantum Graphs: A Review
• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Eigenvalue counting function for Robin Laplacians on conical domains

Cited by in corpus (9)

• Dirichlet spectrum of the Fichera layer
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• Spectral properties of soft quantum waveguides
• Eigenvalue counting function for Robin Laplacians on conical domains
• Attractive conical surfaces create infinitely many bound states
• Spectral asymptotics for $\delta$-interactions on sharp cones
• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions
• 2D Schrödinger operators with singular potentials concentrated near curves
• Existence of the discrete spectrum in the Fichera layers and crosses of arbitrary dimension