On geometric perturbations of critical Schrödinger operators with a surface interaction

Abstract

We study singular Schrödinger operators with an attractive interaction supported by a closed smooth surface A ⊂ R 3 and analyze their behavior in the vicinity of the critical situation where such an operator has empty discrete spectrum and a threshold resonance. In particular, we show that if A is a sphere and the critical coupling is constant over it, any sufficiently small smooth area preserving radial deformation gives rise to isolated eigenvalues. On the other hand, the discrete spectrum may be empty for general deformations. We also derive a related inequality for capacities associated with such surfaces.

Generalized interactions supported on hypersurfaces

“…This is known for δ-interactions [6,13] and the same holds true for the generalized interactions with β = 0 identically, as the following observation shows. …”

A spectral isoperimetric inequality for cones

“…The corresponding problem in three dimensions is more involved. For closed simply connected surfaces of a fixed area the sphere gives a local maximum of the ground-state eigenvalue, however, the result does not have a global validity [17].…”

Quantum Waveguides

“…Another partial analogue of Theorem 10.6 concerns δ-interactions supported by a surface ⊂ R 3 . If the latter is a sphere and the coupling constant α is such that the corresponding operator H α, is critical, then any small area-preserving deformation of gives rise to a non-void discrete spectrum [EFr09]. This result holds only locally, however, there are large deformations which do not produce any eigenvalues.…”