Regularization for the Schrödinger equation with rough potential: one-dimensional case
Ruobing Bai, Yajie Lian, Yifei Wu
TL;DR
This work provides a complete regularity classification for the one-dimensional Schrödinger equation with rough spatial potentials $\eta\in L_x^r+L_x^{\infty}$, revealing sharp global well-posedness thresholds in $H_x^s$ that depend on $r$. By combining commutator estimates, local smoothing, and a normal-form transformation, the authors overcome the inability to differentiate rough potentials and establish sharp regularity up to $H_x^{\tfrac32-}$ for $r=1$, $H_x^{\tfrac52-\tfrac1r}$ for $1<r\le2$, and $H_x^{2}$ for $r>2$, with ill-posedness beyond these limits demonstrated via explicit constructions. The analysis leverages a resonant/non-resonant decomposition to manage frequency interactions, and provides insights into how roughness constrains or augments dispersive regularization in linear and, by extension, nonlinear settings. Additionally, the paper discusses, without proof, the influence of nonlinearity on regularity, and situates the results within the broader context of sharp well-posedness theory for dispersive equations with singular potentials. Overall, the results offer precise guidance for understanding localization, regularity, and numerical approximation in disordered Schrödinger dynamics.
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 $η$ is a given spatial potential (including the delta potential and $|x|^{-γ}$-potential). Our goal is to provide the regularization mechanism of this model when the potential $η\in L_x^r+L_x^\infty$ is rough. In this paper, we mainly focus on one-dimensional case and establish the following results: 1) When the potential $η\in L_x^1+L_x^\infty(\mathbb{R})$, then the solution is in $H_x^{\frac 32-}(\mathbb{R})$; however, there exists some $η\in L_x^1+L_x^\infty(\mathbb{R})$ such that the solution is not in $H_x^{\frac 32}(\mathbb{R})$; 2) When the potential $η\in L_x^r+L_x^\infty(\mathbb{R})$ for $1<r\leq 2$, then the solution is in $H_x^{\frac 52-\frac 1r}(\mathbb{R})$; however, there exists some $η\in L_x^r+L_x^\infty(\mathbb{R})$ such that the solution is not in $H_x^{\frac 52-\frac 1r+}(\mathbb{R})$; 3) When the potential $η\in L_x^r+L_x^\infty(\mathbb{R})$ for $r>2$, then the solution is in $H_x^{2}(\mathbb{R})$; however, there exists some $η\in L_x^r+L_x^\infty(\mathbb{R})$ 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.