Lower bounds on the lowest spectral gap of singular potential Hamiltonians

Abstract

We analyze Schrödinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional Euclidean space the size of the gap depends only on the following parameters: the length, diameter and maximal curvature of the curve, a certain parameter measuring the injectivity of the curve embedding, and a compact sub-interval of the open, negative energy half-axis which contains the two lowest eigenvalues.

arXiv comment24 pages. To appear in slightly different form in Annales Henri Poincare

References in corpus (3)

• Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop
• Exponential splitting of bound states in a waveguide with a pair of distant windows
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Cited by in corpus (9)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc
• Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions
• Asymptotic behaviour of the spectrum of a waveguide with distant perturbations
• Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
• The Allegretto-Piepenbrink Theorem for strongly local forms
• Spectral gaps for self-adjoint second order operators