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

TL;DR

This paper analyzes two-dimensional Schrödinger operators with point interactions, classifying their threshold behaviors, and establishing conditions for zero modes and $L^p$-boundedness of wave operators.

Contribution

It provides a comprehensive classification of the threshold behavior of such operators and establishes $L^p$ boundedness results for wave operators in various cases.

Findings

Operators can be of regular or singular type at threshold

Zero eigenvalues only occur with three or more centers

Wave operators are $L^p$-bounded for all $1<p< { }\infty$ in regular cases

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.