Convergence over fractals for the periodic Schrödinger equation

Abstract

We consider a fractal refinement of the Carleson problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For α ∈ (0, d] and s < d 2(d+1) (d + 1 − α), we find a function in H s (T d ) whose corresponding solution diverges in the limit t → 0 on a set with strictly positive α-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that s > d 2(d+2) (d + 2 − α) is sufficient for the solution corresponding to every datum in H s (T d ) to converge α-almost everywhere.

Convergence over fractals for the Schrödinger equation

“…When n = 1, this result is due to Carleson [7], whose proof extends to higher dimensions as proved, for instance, in [18]. Moreover, we can show that this limit exists γ-almost everywhere for every f ∈ H s with s ∈ (0, n/2], as long as γ > n − 2s; see the appendix of [16]. This can be regarded as a refinement of Carleson's result, although it does not recover it.…”

“…To deal with perturbations of quadratic sums, we will use the following Lemma, which is consequence of Abel's summation formula; see Lemma 2.3 of [16].…”

“…It is worth mentioning that (3) is necessary for the α-a.e. pointwise convergence in the periodic setting [16], however in this setting it is still unknown if it is sufficient (not even for α = n).…”

Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$

“…Actually, this is indicated in the proof of Eceizabarrena and Lucà [25]. Indeed, they show that there exists a function f ∈ H s for s < d 2(d+1) which consists of segmented Gauss sums fails to convergence on a nontrivial Hausdorff dimension set(See Theorem 1.1 of [25]). Here, we explicate the proof of the lower bound of dim H L α via Proposition 4.2.…”

“…To prove the upper bound in Proposition 1.8, we utilize a completion method of higher dimension which is developed by Chen, Shparlinski in [15]. Meanwhile, we use a fractal maximal estimate of the periodic Schrödinger operator derived from Theorem 1.1, which can be obtained by the argument of Eceizabarrena and Lucà in [25]. Barron in [4] studied the similar problem in one dimension.…”