An Upbound of Hausdorff's Dimension of the Divergence Set of the fractional Schr\"odinger Operator on $H^s(\mathbb R^n)

TL;DR

This paper investigates the Hausdorff dimension of the divergence set for the fractional Schrödinger operator on Sobolev spaces, establishing an upper bound under specific conditions on the parameters.

Contribution

It provides a new upper bound on the Hausdorff dimension of divergence sets for fractional Schrödinger evolutions in Sobolev spaces, extending previous results to fractional orders.

Findings

Upper bound on divergence set dimension derived

Conditions on parameters for the bound established

Results applicable to fractional Schrödinger operators with lpha > 1/2

Abstract

This paper shows $$ \sup_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{it(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}\ \ \text{under}\ \ \begin{cases} n\geq2;\\ \alpha>\frac12; \frac{n}{2(n+1)}<s\leq\frac{n}{2} . \end{cases} $$