On the Dirac delta as initial condition for nonlinear Schr\"odinger equations

TL;DR

This paper investigates the initial value problem for nonlinear Schr"odinger equations with Dirac delta initial data, demonstrating finite energy solutions after renormalization in certain cases and establishing stability results.

Contribution

It introduces a novel analysis of Schr"odinger equations with singular initial data, including renormalization techniques and stability results in the defocusing case.

Findings

Finite energy solutions after renormalization in the critical one-dimensional case.

Stability results for solutions with Dirac delta initial data in the defocusing setting.

Connection to singular dynamics of Schr"odinger maps via Hasimoto transformation.

Abstract

In this article we will study the initial value problem for some Schr\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schr\"odinger maps through the so called Hasimoto transformation.