High frequency dispersive estimates for the Schrodinger equation in high dimensions

TL;DR

This paper establishes optimal high-frequency dispersive estimates for the Schrödinger equation in dimensions four and higher with certain decaying potentials, advancing understanding of wave behavior in high-dimensional quantum systems.

Contribution

It provides new optimal dispersive estimates for the Schrödinger group with specific decay and regularity conditions on the potential in high dimensions.

Findings

Proves optimal dispersive estimates for high-frequency Schrödinger group

Identifies conditions on potential decay and regularity for estimates

Offers a sufficient condition related to $L^1\to L^\infty$ bounds for potential iteration

Abstract

We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(|x|^{-\delta})$, $\delta>n-1$, and $V\in C^k({\bf R}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)/2\le k_n<n/2$. We also give a sufficient condition in terms of $L^1\to L^\infty$ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.