Restriction Theorems and Strichartz Inequalities for the Laguerre Operator Involving Orthonormal Functions

In this paper, we prove restriction theorems for the Fourier–Laguerre transform and establish Strichartz estimates for the Schrödinger propagator \(e^{-itL_\alpha }\) $e^{-it{L}_{\alpha }}$ for the Laguerre operator \(L_\alpha =-\Delta -\sum _{j=1}^{n}(\dfrac{2\alpha _j+1}{x_j}\dfrac{\partial }{\partial x_j})+\dfrac{|x|^2}{4}\) $L_{\alpha }=-\mathrm{\Delta }-{\sum }_{j=1}^n({\displaystyle \frac{2{\alpha }_j+1}{x_j}}{\displaystyle \frac{\partial }{\partial x_j}})+{\displaystyle \frac{{|x|}^2}{4}}$ , \(\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)\in {(-\frac{1}{2},\infty )^n}\) $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\in {(-\frac{1}{2},\infty )}^n$ on \(\mathbb {R}_+^n\) ${\mathbb{R}}_{+}^n$ involving systems of orthonormal functions. The proof is based on a combination of some known dispersive estimate and the argument in Nakamura [Trans Am Math Soc 373(2), 1455–1476 (2020)] on torus. As an application, we obtain the global well-posedness for the nonlinear Laguerre–Hartree equation in Schatten space.