Spectral asymptotics for $\delta$-interactions on sharp cones

Abstract

We investigate the spectrum of three-dimensional Schrödinger operators with $\delta$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions.

References in corpus (4)

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

Cited by in corpus (2)

• Optimization of the lowest eigenvalue of a soft quantum ring
• Attractive conical surfaces create infinitely many bound states