On the dimension of divergence sets of Schr\"odinger equation with complex time

Abstract

This article studies the pointwise convergence for the fractional Schr\"odinger operator $P^{t}_{a,\gamma}$ with complex time in one spatial dimension. Through establishing $L^2$-maximal estimates for initial datum in $H^{s}(\mathbb{R})$, we see that the solution converges to the initial data almost everywhere with $s>\frac14 a(1-\frac1\gamma)_+$ when $0<a<1$ and $s>\frac{1}{2}(1-\frac{1}{\gamma})_{+}$ when $a=1$. By constructing counterexamples, we show that this result is almost sharp up to the endpoint. These results extends the results of P. Sj\"olin, F. Soria and A. Baily. Second, we study the Hausdorff dimension of the set of the divergent points, by showing some $L^1$-maximal estimates with respect to general Borel measure. Our results reflect the interaction between dispersion effect and dissipation effect, arising from the fractional Schr\"{o}dinger type operator $P^{t}_{a,\gamma}$ with the complex time.