# Dispersive estimates for Schr\"odinger operators with point interactions in $\mathbb{R}^3$

## TL;DR

This paper establishes dispersive $L^p-L^q$ estimates for Schr"odinger operators with a single point interaction in three dimensions, extending understanding of quantum systems with singular short-range potentials.

## Contribution

It provides a direct and simple proof that the perturbed Laplacian with one point interaction satisfies certain dispersive estimates, matching the free case in a specific regime.

## Key findings

- Dispersive estimates hold for $q ext{ in }[2,3)$
- The approach is more direct than previous methods
- Potential to extend estimates to $q ext{ greater or equal to }3$

## Abstract

The study of dispersive properties of Schr\"odinger 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 Schr\"odinger 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 $q\geq 3$.