Convergence over fractals for the periodic Schrödinger equation

Abstract

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For $\alpha \in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - α), \] we find a function in $H^s({\mathbb{T}}^d)$ whose corresponding solution diverges in the limit $t\to 0$ on a set with strictly positive $\alpha$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that \[ s > \frac{d}{2(d+2)}\left( d+2-α\right) \] is sufficient for the solution corresponding to every datum in $H^s({\mathbb{T}}^d)$ to converge to such datum $\alpha$-almost everywhere.

v3: Accepted manuscript

References in corpus (2)

• Riemann's non-differentiable function and the binormal curvature flow
• Geometric differentiability of Riemann's non-differentiable function

Cited by in corpus (5)

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• Convergence over fractals for the Schrödinger equation
• Pointwise convergence of the Klein-Gordon flow
• Pointwise convergence problem of Ostrovsky equation with rough data and random data