Sharp pointwise convergence of Schr\"odinger mean with complex time in higher dimensions

TL;DR

This paper proves that solutions to the Schrödinger equation with complex time converge almost everywhere in higher dimensions, given initial data in a Sobolev space, advancing understanding of complex-time Schrödinger dynamics.

Contribution

It establishes the pointwise convergence of Schrödinger solutions with complex time in higher dimensions, a novel result in the analysis of complex-time evolution.

Findings

Almost everywhere convergence of Schrödinger solutions with complex time

Convergence holds for initial data in Sobolev space $H^s(\mathbb{R}^d)$

Extension of convergence results to higher dimensions

Abstract

In this paper, we establish the almost everywhere convergence of solutions to the Schr\"odinger operator with complex time $ P_{\gamma}f(x,t) $ in higher dimensions, under the assumption that the initial data $f$ belongs to the Sobolev space $ H^{s}(\mathbb{R}^d)$.