Two-dimensional Schrödinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operators

We study the threshold behavior of two-dimensional Schrödinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with [Formula: see text] or [Formula: see text]-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centers. When the operator is of regular type, we prove that the wave operators are bounded in [Formula: see text] for all [Formula: see text]. With a single center, we always are in the regular type case.

Wigner–von Neumann perturbations of a periodic potential: spectral singularities in bands

Wigner–von Neumann type perturbations of the periodic one-dimensional Schrödinger operator are considered. The asymptotics of the solution to the generalized eigenfunction equation is investigated. It is proven that a subordinated solution and therefore an embedded eigenvalue may occur at the points of the absolutely continuous spectrum satisfying a certain resonance (quantization) condition between the frequencies of the perturbation, the frequency of the background potential and the corresponding quasimomentum.

METASTABLE STATES IN A MODEL OF NUCLEAR PAIRING

We consider a simplified linearized quantum mechanical description of pairing in atomic nuclei. The model is given by the Hamiltonian: Hε= H(0)+εH(1), where H(j), for j=0,1, are two 2×2 symmetric operator valued matrices, and ε>0. Physically, H(0) and H(1) represent respectively the density matrix and the pairing contribution to the Hamiltonian. We study this model for small ε. If E0 is an embedded eigenvalue of H0, then the corresponding eigenstate Φ becomes unstable when ε≠0, and we give an asymptotic estimate for the time evolution of Φ under Hε, in terms of the coupling strength ε. We illustrate this result on the case of a solvable model with point interaction, and show that the above asymptotic result is, at least for some matrix elements, almost optimal.