Uniqueness of solutions to nonlinear Schr\"odinger equations from their zeros

TL;DR

This paper establishes new uniqueness results for solutions to nonlinear Schr"odinger equations, showing that under certain zero conditions on subsets of the domain, the only solution is trivial, extending classical Fourier uniqueness concepts.

Contribution

It introduces novel uniqueness and rigidity theorems for nonlinear Schr"odinger equations with zero conditions on specific subsets, linking to Fourier uniqueness and elliptic equations.

Findings

The trivial solution is unique under zero conditions on certain subsets.

Results extend Fourier uniqueness pairs to nonlinear PDEs.

Rigidity results for semilinear elliptic equations from zeros.

Abstract

We show novel types of uniqueness and rigidity results for Schr\"odinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only solution for which the assumptions $u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$ hold, where $D\subset \mathbb{R}^d$ are certain subsets of codimension one. In particular, $D$ is discrete for dimension $d=1$. Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko--Viazovska formula, and the uniqueness result of the second author and M. Sousa for powers of integers. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros.