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

TL;DR

This paper extends the understanding of regularization mechanisms for the Schr"odinger equation with rough potentials to high-dimensional spaces, identifying conditions for well-posedness and optimal regularity based on potential roughness.

Contribution

It provides a comprehensive high-dimensional analysis of Schr"odinger equations with rough potentials, establishing well-posedness thresholds and regularity characterizations.

Findings

Ill-posedness for $r<d/2$ in high dimensions.

Optimal regularity $H_x^{ ext{min}igrace{2+d/2 - d/r, 2igrace}}$ for $r o d/2$ and above.

Characterization of regularity with a notable exception at $d=2, r=1$.

Abstract

In this work, we investigate the regularization mechanisms of the Schr\"odinger equation with a spatial potential $$ i\partial_t u+\Delta u+\eta u =0, $$ where $\eta$ denotes a given spatial potential. The regularity of solutions constitutes one of the central problems in the theory of dispersive equations. Recent works \cite{Bai-Lian-Wu-2024, M-Wu-Z24} have established the sharp regularization mechanisms for this model in the whole space $\mathbb{R}$ and on the torus $\mathbb{T}$, with $\eta$ being a rough potential. The present paper extends the line of research to the high-dimensional setting with rough potentials $\eta \in L_x^r+L_x^{\infty}$. More precisely, we first show that when $1\leq r <\frac d2$, there exists some $\eta \in L_x^r+L_x^{\infty}$ such that the equation is ill-posed in $H_x^{\gamma}$ for any $\gamma \in \mathbb{R}$. Conversely, when $\frac d2 \leq r \leq \infty$, the expected optimal regularity is given by $$H_x^{\gamma_*}, \quad \gamma_*=\mbox{min}\{2+\frac d2-\frac dr, 2\}.$$ We establish a comprehensive characterization of the regularity, with the exception of two dimensional endpoint case $d=2, r=1$. Our novel theoretical framework combines several fundamental ingredients: the construction of counterexamples, the proposal of splitting normal form method, and the iterative Duhamel construction. Furthermore, we briefly discuss the effect of the interaction between rough potentials and nonlinear terms on the regularity of solutions.