On the dimension of divergence sets of Schrödinger equation with complex time

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