Rainer Hempel

On eigenvalues in gaps for perturbed magnetic Schrödinger operators

We study Schrodinger operators H0 with a gap in the essential spectrum, perturbed by either a decreasing electric potential or a decreasing magnetic field; in both cases the strength of the perturbation is measured by a coupling constant λ⩾0. Here we are mainly interested in the asymptotic behavior (as λ→∞) of certain counting functions for the eigenvalues that are produced by the perturbation inside the spectral gap. The case where we perturb by a potential can be handled using current technology, even if H0 contains a fixed magnetic background. For perturbations by magnetic fields, however, we require rather strong assumptions—like exponential decay of the perturbations—to obtain a lower bound on the counting function. To gain some additional intuition, we use separation of variables in the closely related model of a Schrodinger operator with constant magnetic field in R2, perturbed by a rotationally symmetric magnetic field that decays at infinity.

ON OPEN SCATTERING CHANNELS FOR MANIFOLDS WITH ENDS

Abstract In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension n. The smallness condition for the perturbation is expressed (intrinsically and coordinate free) in purely geometric terms using the harmonic radius; therefore, the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the Ricci-curvature. As an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the Riemannian metric. This stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations.

On the Asymptotic Distribution of Eigenvalues in Gaps

Virtually all results on eigenvalue asymptotics for differential operators have their roots in Weyl’s celebrated law for the distribution of the eigenvalues

Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries

0 < E_1 < E_2 \leqslant E_3 \leqslant \ldots ,E_k \to \infty {\text{ }}as{\text{ }}k \to \infty ,

Spectral properties of grain boundaries at small angles of rotation

of the Dirichlet Laplacian -△ on an open, bounded domain Ω ⊂ R m : If N(λ) denotes the number of eigenvalues E k < λ, then

Discrete and Cantor Spectrum for Neumann Laplacians of Combs

N\left( \lambda \right) \sim c_d vol\left( \Omega \right)\lambda ^{m/2} ,\lambda \to \infty ,