Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces

Abstract

The main objective of this paper is to investigate orthonormal Strichartz estimates for Schrödinger equation on the Wiener amalgam space $\mathcal{W}(\mathcal{F} L^p, L^q)$. More precisely, we first examine improvements in the time-integrability exponent of the existing Strichartz estimates established by Cordero and Nicola in \cite{NFC} for Schrödinger equations associated with $\mathcal{H}^{+}=-\frac{1}{4π}Δ+|x|^2$. We then extend these improved Strichartz estimates from a single initial datum to systems of orthonormal families, providing, to the best of our knowledge, the first results in this direction in the setting of Wiener amalgam spaces. We further extend the classical Strichartz estimates in Wiener amalgam spaces from a single initial datum to systems of orthonormal families of initial data for the Schrödinger equation associated with the operator $\mathcal{H}^{-}=-\frac{1}{4π}Δ-|x|^2$ and Hamiltonian operator of the form $\mathcal{H}_{\mathcal{A}} = -\frac{1}{4π} B \nabla \cdot \nabla$, where $\mathcal{A} = \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \in \mathrm{Sp}(d,\mathbb{R})$ with $B = B^*$ and $\det B \neq0$. A key ingredient of our approach is Stein's complex interpolation theory for Wiener amalgam spaces, combined with a duality argument inspired by the work of Frank and Sabin. As an application of these orthonormal estimates associated with $\mathcal{H}^{+}, \mathcal{H}^{-}$, and $H_{\mathcal{A}},$ we establish local and small-data global well-posedness for the Hartree equation with infinitely many particles, for non-trace-class initial data.