Regularization for the Schrödinger equation with rough potential: one-dimensional case

Abstract

In this work, we investigate the following Schrödinger equation with a spatial potential \begin{align*} i\partial_t u+\partial_x^2 u+ηu=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 $H_x^{\frac{3}{2}-}(\mathbb{R})$; however, there exists some $\eta \in L_x^1+L_x^{\infty }(\mathbb{R})$ such that the solution is not in $H_x^{\frac{3}{2}}(\mathbb{R})$; 2) When the potential $\eta \in L_x^r+L_x^{\infty }(\mathbb{R})$ for $1<r\le 2$, then the solution is in $H_x^{\frac{5}{2}-\frac{1}{r}}(\mathbb{R})$; however, there exists some $\eta \in L_x^r+L_x^{\infty }(\mathbb{R})$ such that the solution is not in $H_x^{\frac{5}{2}-\frac{1}{r}+}(\mathbb{R})$; 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. Moreover, a vector-valued Coifman-Meyer multiplier theorem is established, which extends the classical multi-linear multiplier theory to the time-space mixed norm setting. Additionally, we also discuss, without proof, the influence of the existence of nonlinearity on the regularity of solution.