The orthonormal Strichartz estimates and convergence problem of density functions related to ${\mathrm{\partial }}_x^3+{\mathrm{\partial }}_x^{-1}$

In this article, we investigate the orthonormal Strichartz estimates and the convergence problem of the density function associated with ${\mathrm{\partial }}_x^3+{\mathrm{\partial }}_x^{-1}$. Firstly, when ${\gamma }_0\in {\mathfrak{S}}^{\beta }({\dot{H}}^s)$ with $\frac{1}{4}\le s<\frac{1}{2},0<\alpha \le 1$, and $1\le \beta <\frac{\alpha }{1-2s}$, we prove that $\underset{t⟶0}{\lim }{\sum }_{j=1}^{+\mathrm{\infty }}{\lambda }_j{\mid e^{-t({\mathrm{\partial }}_x^3+{\mathrm{\partial }}_x^{-1})}{f}_j\mid }^2={\sum }_{j=1}^{+\mathrm{\infty }}{\lambda }_j{\mid f_j\mid }^2\mathrm{.}$ This extends the Theorem 1.1 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). Moreover, we present the Hausdorff dimension of the divergence set of the density function related to ${\mathrm{\partial }}_x^3+{\mathrm{\partial }}_x^{-1}$, namely $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}D({\gamma }_0)\le (1-2s)\beta$, which extends the Theorem 1.1 of Zhao et al. (Acta Math. Sci. Ser. B (Engl. Ed.) 42(2022), 1607-1620.). % Secondly, we present the orthonormal Strichartz estimates and the Schatten bounds with space-time norms on $\mathbf{R}$. % Finally, by using full randomization, we establish the probabilistic convergence of the density function related to ${\mathrm{\partial }}_x^3+{\mathrm{\partial }}_x^{-1}$ on $\mathbb{R}$, which extends the Theorem 1.3 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.).