Dispersive estimates for Schrödinger operators with point interactions in ${\mathbb{R}}^3$

Abstract

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 $L^p$ boundedness 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 $q\ge 3$.

To appear on: "Advances in Quantum Mechanics: Contemporary Trends and Open Problems", G. Dell'Antonio and A. Michelangeli eds., Springer-INdAM series 2017

References in corpus (1)

• On wave operators for Schrödinger operators with threshold singuralities in three dimensions

Cited by in corpus (1)

• The $L^p$ boundedness of wave operators for the three-dimensional multi-centre point interaction