Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol

Abstract

We study the problem of pointwise convergence for equations of the type $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-\alpha+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^\alpha$-a.e, where $\mathcal{H}^\alpha$ is the $\alpha$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of the symbol. On the other hand, we prove negative results for a wide family of polynomial symbols $P$. Given $\alpha$, we exploit a Talbot-like effect to construct regular initial data whose solutions $u$ diverge in sets of Hausdorff dimension $\alpha$. However, for quadratic symbols like the saddle, other kind of examples show that our positive results are sometimes best possible. To compute the dimension of the sets of divergence we use a Mass Transference Principle from Diophantine approximation theory