QUARTERLY OF APPLIED MATHEMATICS VOLUME LXXV, NUMBER 3 SEPTEMBER 2017, PAGES 555–579 http://dx.doi.org/10.1090/qam/1465 Article electronically published on March 6, 2017 DYNAMICAL SYSTEM APPROACH TO SYNCHRONIZATION OF THE COUPLED SCHR ¨ ODINGER–LOHE SYSTEM By HYUNGJIN HUH (Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea ) and SEUNG-YEAL HA (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 ) Abstract. We study wave function synchronization of the Schr¨odinger–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. 1. Introduction. The collective synchronous behaviors of classical complex systems are ubiquitous in nature, e.g., the flashing of fireflies, clapping of hands in a concert hall, and heartbeat regulation by pacemaker cells [1–3,24,25]. However, rigorous mathematical studies of these collective phenomena were performed only several decades ago by Winfree Received February 6, 2017. 2010 Mathematics Subject Classification. Primary 82C10; Secondary 82C22, 35B35. Key words and phrases. Complete synchronization, quantum synchronization, Schr¨ odinger–Lohe model. 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. E-mail address : huh@cau.ac.kr E-mail address : syha@snu.ac.kr c 2017 Brown University 555