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.