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Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3

TitleDispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3
Publication TypeBook Chapter
Year of Publication2017
AuthorsIandoli, F, Scandone, R
EditorMichelangeli, A, Dell'Antonio, G
Book TitleAdvances in Quantum Mechanics: Contemporary Trends and Open Problems
PublisherSpringer International Publishing
ISBN Number978-3-319-58904-6

The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q \geq 3$.


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