Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system

Authors: Hyungjin Huh and Seung-Yeal Ha
Journal: Quart. Appl. Math. 75 (2017), 555-579
MSC (2010): Primary 82C10; Secondary 82C22, 35B35
DOI: https://doi.org/10.1090/qam/1465
Published electronically: March 6, 2017
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Abstract: We study wave function synchronization of the Schrödinger-Lohe model, which describes the dynamics of the ensemble of coupled quantum Lohe oscillators with infinite states. To do this, we first derive a coupled system of ordinary differential equations for the $ L^2_x$ inner products between distinct wave functions. For the same one-body potentials, we show that the inner products of two wave functions converge to unity for some restricted class of initial data, so complete wave function synchronization emerges asymptotically when the dynamical system approach is used. Moreover, for the family of one-body potentials consisting of real-value translations of the same base potential, we show that the inner products for a two-oscillator system follow the motion of harmonic oscillators in a small coupling regime, and then as the coupling strength increases, the inner products converge to constant values; this behavior yields convergence toward constant values for the $ L^2_x$ differences between distinct wave functions.

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Additional Information

Hyungjin Huh
Affiliation: Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
Email: huh@cau.ac.kr

Seung-Yeal Ha
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea – and – Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Korea
Email: syha@snu.ac.kr

DOI: https://doi.org/10.1090/qam/1465
Keywords: Complete synchronization, quantum synchronization, Schr\"odinger--Lohe model
Received by editor(s): February 6, 2017
Published electronically: March 6, 2017
Additional Notes: The first author was supported in part by the Basic Science Research Program through the NRF funded by the Ministry of Education (2014R1A1A2053747)
The work of the second author was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03
Article copyright: © Copyright 2017 Brown University