Sobolev norms of $L^2$-solutions to NLS

TL;DR

This paper uses inverse spectral theory to identify conserved quantities for solutions of the nonlinear Schrödinger equation, linking them to Sobolev norms in a specific range of regularity.

Contribution

It introduces a novel approach to connect spectral invariants with Sobolev norms for NLS solutions with low regularity initial data.

Findings

Existence of conserved quantities equivalent to Sobolev norms for NLS

Extension of spectral methods to low-regularity initial data

Identification of invariants for solutions in H^s with s in [-1,0]

Abstract

We apply inverse spectral theory to study Sobolev norms of solutions to the nonlinear Schrodinger equation. For initial datum $q_0\in L^2(\mathbb{R})$ and $s\in [-1,0]$, we prove that there exists a conserved quantity that is equivalent to $H^s(\mathbb{R})$-norm of the solution.