Regularization for the Schr\"{o}dinger equation with rough potential: one-dimensional case

TL;DR

This paper studies how rough spatial potentials affect the regularity of solutions to the one-dimensional Schrödinger equation, establishing precise regularity thresholds depending on the potential's integrability class.

Contribution

It provides a complete classification of solution regularity for the Schrödinger equation with rough potentials in one dimension, using commutator, smoothing, and normal form techniques.

Findings

Solutions gain regularity depending on the potential's integrability class.

Existence of potentials where solutions do not reach the expected regularity.

Complete characterization of regularity thresholds for different potential classes.

Abstract

In this work, we investigate the following Schr\"odinger equation with a spatial potential \begin{align*} i\partial_t u+\partial_x^2 u+\eta u=0, \end{align*} where $\eta$ is a given spatial potential (including the delta potential and $|x|^{-\gamma}$-potential). Our goal is to provide the regularization mechanism of this model when the potential $\eta\in L_x^r+L_x^\infty$ is rough. In this paper, we mainly focus on one-dimensional case and establish the following results: 1) When the potential $\eta \in L_x^1+L_x^\infty(\mathbb{R})$, then the solution is in $H_x^{\frac 32-}(\mathbb{R})$; however, there exists some $\eta \in L_x^1+L_x^\infty(\mathbb{R})$ such that the solution is not in $H_x^{\frac 32}(\mathbb{R})$; 2) When the potential $\eta \in L_x^r+L_x^\infty(\mathbb{R})$ for $1<r\leq 2$, then the solution is in $H_x^{\frac 52-\frac 1r}(\mathbb{R})$; however, there exists some $\eta \in L_x^r+L_x^\infty(\mathbb{R})$ such that the solution is not in $H_x^{\frac 52-\frac 1r+}(\mathbb{R})$; 3) When the potential $\eta \in L_x^r+L_x^\infty(\mathbb{R})$ for $r>2$, then the solution is in $H_x^{2}(\mathbb{R})$; however, there exists some $\eta \in L_x^r+L_x^\infty(\mathbb{R})$ such that the solution is not in $H_x^{2+}(\mathbb{R})$. Hence, we provide a complete classification of the regularity mechanism. Our proof is mainly based on the application of the commutator, local smoothing effect and normal form method. Additionally, we also discuss, without proof, the influence of the existence of nonlinearity on the regularity of solution.