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

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.