Higher-dimensional orthonormal Strichartz estimates and expanded exponent ranges

Establish orthonormal Strichartz inequalities for the anisotropic Hamiltonian \(\mathcal{H}_{\mathcal{A}}=-\frac{1}{4\pi}B\nabla\cdot\nabla\) and the inverted harmonic oscillator \(\mathcal{H}^{-}=-\frac{1}{4\pi}\Delta-\pi|x|^2\) in dimensions higher than one, and determine whether the admissible range of the exponent \(p\) in the corresponding one-dimensional estimates can be improved.

Background

The paper proves orthonormal Strichartz estimates for the anisotropic Hamiltonian $\mathcal{H}_{\mathcal{A}}$ and the inverted harmonic oscillator $\mathcal{H}^{-}$ only in dimension $d=1$. The authors explicitly ask whether analogous inequalities can be established in higher dimensions.

The authors also indicate that their use of weak kernel decay may have produced a non-sharp range of admissible spatial exponents $p$. Determining the optimal range would strengthen Theorems $\ref{theoremONSHA}$ and $\ref{theoremONSH-}$.