Local decay estimates for the bi-Laplacian Nonautonomous Schr\"{o}dinger equation

TL;DR

This paper establishes local decay estimates for the bi-Laplacian Schrödinger equation with time-dependent potentials in high dimensions, extending previous results and deriving global Strichartz estimates.

Contribution

It introduces a novel approach based on asymptotic completeness and channel wave operators to obtain decay estimates, surpassing standard resolvent methods.

Findings

Local decay estimates proven for dimensions n≥14 and n≥9 with spectral regularity.

Extension of decay results to quasi-periodic potentials.

Derivation of global-in-time Strichartz estimates from local decay.

Abstract

In this paper, we establish local decay estimates for the bi-Laplacian Schr\"{o}dinger equation with time-dependent (in particular, quasi-periodic) potentials in spatial dimension $n\ge14$. Moreover, under stronger spectral regularity hypotheses, the same result can be extended to dimension $n\ge9$. Our approach, based on asymptotic completeness and the existence of the channel wave operator, departs from standard resolvent-based methods. In addition, global-in-time Strichartz estimates are derived from the local decay estimates.