Lower Bounds on the Lowest Spectral Gap of Singular Potential Hamiltonians

We analyze Schr?dinger 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. Dedicated to Krešimir Veselić on the occasion of his 65^th birthday. Submitted: February 20, 2006; Accepted: May 8, 2006     

Spectral Gap of Segments of Periodic Waveguides

The lowest spectral gap of segments of a periodic waveguide in is proportional to the square of the inverse length. Dedicated to Pavel Exner on the occasion of his 60th birthday.     

Aharonov and Bohm versus Welsh eigenvalues

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the \(\delta \) type and a fixed strength \(\beta \) supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux \(\alpha \in [0,\frac{1}{2}]\) in the center. It is shown that if \(\beta \ne 0\), there is a critical value \(\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})\) such that the discrete spectrum has an accumulation point when \(\alpha <\alpha _{\mathrm {crit}}\), while for \(\alpha \ge \alpha _{\mathrm {crit}}\) the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed \(\alpha \in (0,\frac{1}{2})\) and \(|\beta |\) small enough.     

Resonances Induced by Broken Symmetry in a System with a Singular Potential

We study a three-dimensional nonrelativistic quantum system with a delta-type potential. The support of this potential is determined by a circle and straight line in ${\mathbb{R}^3}$ . We show that a special symmetry of our system induces embedded eigenvalues, and breaking this symmetry leads to resonances.     

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

Letters in Mathematical Physics - We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or...     

On the Eigenvalue Problem for Self–Adjoint Operators with Singular Perturbations

We investigate the eigenvalue problem for self–adjoint operators with singular perturbations. The general results presented here include weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so–called additive strongly singular perturbations.