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

DOI10.1142/S0129055X19500120zbMATH Open1423.35274arXiv1804.01297OpenAlexW2795877358MaRDI QIDQ5237786

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Publication date: 18 October 2019

Published in: (Search for Journal in Brave)

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 $L^p(R^2)$ for all $1<p<infty$. With a single center we always are in the regular type case.

Full work available at URL: https://arxiv.org/abs/1804.01297

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