Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators

Abstract

We study the threshold behaviour of two dimensional Schr{\" o}dinger 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 $s$ or p-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 centres. When the operator is of regular type we prove that the wave operators are bounded in $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$. With a single center we always are in the regular type case.

arXiv comment26 pages

References in corpus (5)

• Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy
• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
• Wave operator bounds for 1-dimensional Schrödinger operators with singular potentials and applications
• The $L^p$ boundedness of wave operators for the three-dimensional multi-centre point interaction
• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Cited by in corpus (9)

• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
• $L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions
• On the approximation by regular potentials of Schrödinger operators with point interactions
• Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting
• On real resonances for three-dimensional Schrödinger operators with point interactions
• On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction
• Scattering operator and wave operators for 2D Schrödinger operators with threshold obstructions
• Resonances and resonance expansions for point interactions on the half-space
• Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction