The convergence rate for Schr\

Tie Li; Yaoming Niu; Ying Xue

doi:10.4134/JKMS.j230461

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2024; 61(6): 1145-1170

Online first article October 18, 2024 Printed November 1, 2024

https://doi.org/10.4134/JKMS.j230461

The convergence rate for Schr\"{o}dinger operators with complex time

Tie Li, Yaoming Niu, Ying Xue

Inner Mongolia University of Science and Technology; Ordos Institute of Technology; Inner Mongolia University of Science and Technology

Abstract

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Abstract

In this paper, in one spatial dimension, we study the convergence rate for Schr\"{o}dinger operators with complex time $P_{a,\gamma}^{t}$, which is defined by $$P_{a,\gamma}^{t}f(x)=S_{a}^{t+it^{\gamma}}f(x) =\int_{\mathbb{R}} e^{ix\xi}e^{it|\xi|^{a}}e^{-t^{\gamma}|\xi|^{a}} \hat{f}(\xi)d\xi,$$ where $\gamma>0$ and $a>0.$ The convergence rate for Schr\"{o}dinger operators with complex time is different from that of classical Schr\"{o}dinger operators in Cao-Fan-Wang (Illinois J. Math. 62: 365--380, 2018).

Keywords: Schr\"{o}dinger operator, maximal operator, complex time, local estimate, global estimate

MSC numbers: Primary 42B25; Secondary 35Q55

Supported by: The work is supported by the National Natural Science Foundation of China (No. 11661061) and by the Natural Science Foundation of Inner Mongolia (No. 2024MS01005), Inner Mongolia University scientific research projects (Nos. NJZY22215, NJZZ21050, NJZY19186), Baotou Teachers' College scientific research project (No. BSYKJ2021-ZQ05).