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 -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.

Keywords:

• Schrödinger operator
• Dirichlet Laplacian
• conical surface
• eigenvalue

AMS Subject Classifications:

• 35J05
• 35P20
• 35J20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Thomas Ourmières-Bonafos is supported by a public grant as part of the “Investissement d’avenir” project, reference [ANR-11-LABX-0056-LMH], LabEx LMH.