A spectral isoperimetric inequality for cones

Abstract

In this note we investigate three-dimensional Schrödinger operators with $\delta$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.

13 pages, revised, to appear in Lett. Math. Phys

References in corpus (5)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Generalized interactions supported on hypersurfaces
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• 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
• An optimization problem for finite point interaction families