Dispersive estimates for higher order Schr\"odinger operators with scaling-critical potentials

TL;DR

This paper establishes dispersive estimates for higher order Schrödinger operators with critical scaling potentials, extending understanding of their time decay properties and spectral behavior in various dimensions.

Contribution

It introduces new dispersive estimates for higher order Schrödinger equations with scaling-critical potentials, including the absence of positive resonances and estimates for modified operators.

Findings

Proves $|t|^{-n/2m}$ decay bounds for the evolution operator.

Shows absence of positive resonances for $(- riangle)^m + V$.

Provides dispersive estimates for operators involving fractional powers of $H$.

Abstract

We prove a family of dispersive estimates for the higher order Schr\"odinger equation $iu_t=(-\Delta)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that $e^{-itH}P_{ac}(H)$ satisfies a $|t|^{-\frac{n}{2m}}$ bound mapping $L^1$ to $L^\infty$ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator $(-\Delta)^m +V$ and a family of dispersive estimates for operators of the form $|H|^{\beta-\frac{n}{2m}}e^{-itH}P_{ac}(H)$ for $0<\beta\leq \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range.