Restriction Theorem and Strichartz estimate for orthonormal functions associated with the Special Hermite Operator
Sunit Ghosh, Jitendriya Swain
TL;DR
The paper addresses Strichartz-type estimates for orthonormal systems associated with the special Hermite operator $\mathcal{L}$ on $\mathbb{C}^n$ for general dispersive flows $e^{-it\phi(\mathcal{L})}$. It develops local restriction estimates for the Fourier–special Hermite transform on surfaces $\{\lambda=\phi(2|\nu|+n)\}$ and translates them into Schatten-space bounds and orthonormal Strichartz inequalities, including extensions to wave, Klein–Gordon, and fractional Schrödinger equations. It establishes a duality principle and proves localized and global Strichartz bounds for orthonormal systems, while showing the endpoint fails, and it derives restriction theorems with Schatten bounds for special Hermite spectral projections, including optimality aspects and local refinements. These results advance understanding of many-body quantum dynamics in the twisted setting and provide sharp spectral-projection estimates with potential applications to dispersive PDEs in complex-valued spaces.
Abstract
Let $\mathcal{L}$ be the special Hermite operator on $\mathbb{C}^n$. As a continuation of the recent results in \cite{SG}, we establish new Strichartz estimates for systems of orthonormal functions associated with general flows of the form $e^{-itφ(\mathcal{L})}$, where $ φ: \mathbb{R}^{+} \to \mathbb{R} $ is a smooth function. Our approach relies on restriction estimates for the Fourier-special Hermite transform on the class of surfaces $\{(λ, μ, ν)\in \mathbb{R}\times\mathbb{N}_0^n\times\mathbb{N}_0^n : λ=φ(2|ν|+n)\}$. We also discuss the endpoint case of the orthonormal Strichartz estimate for the Schrödinger propagator $e^{-it\mathcal{L}}$. Furthermore, we generalize restriction estimates for the special Hermite spectral projections in the context of trace ideals (Schatten spaces).