Orthonormal Strichartz estimates on torus and waveguide manifold and applications

TL;DR

This paper proves new orthonormal Strichartz estimates for fractional Schrödinger equations on torus and waveguide manifolds, extending previous results and introducing novel kernel estimates and decoupling inequalities.

Contribution

It generalizes existing Strichartz estimates to waveguide manifolds and develops new kernel and decoupling estimates for fractional Schrödinger equations.

Findings

Established orthonormal Strichartz estimates on torus and waveguide manifolds.

Derived new kernel estimates generalizing classical dispersive estimates.

Proved local and global well-posedness for Hartree equations with non-trace class initial data.

Abstract

We establish new orthonormal Strichartz estimates for the fractional Schr\"odinger equations on torus $\mathbb T$ and waveguide manifold $\mathbb R^n\times \mathbb T^m$. We generalizes the result of Nakamura [42] on torus; while this is the first result on the waveguide manifold. The main novelty in this paper is the derivation of various kernel estimates associated to the fractional Schr\"odinger equations. Our kernel estimate generalizes the classical dispersive estimate on torus due to Kenig-Ponce-Vega [35]. On the other hand, we obtain new $\ell^2$ decoupling inequality for degeneracy type surfaces to treat the case of waveguide manifold; which maybe of independent interest and complements several known results. As an application, we establish local and small data global well-posednes for the Hartree equation with infinitely many particles with non-trace class initial data.