Quantum synchronization of the Schrödinger-Lohe model (Q2921135)

The authors prove quantum synchronization estimates for the Schrödinger--Lohe model introduced by \textit{M. A. Lohe} [J. Phys. A, Math. Theor. 43, No. 46, Article ID 465301, 20 p. (2010; Zbl 1204.81039)] and which is written as \(i\partial _{t}\psi _{i}=-\Delta \psi _{i}+\frac{K}{N} \sum_{k=1}^{N}(\psi _{k}-\frac{\left\langle \psi _{k},\psi _{i}\right\rangle }{\left\langle \psi _{i},\psi _{i}\right\rangle }\psi _{i})\), \(i=1,\ldots ,N\) , where \(\psi _{i}\) is the wave function of the identical quantum oscillator at the \(i\)th node. Initial conditions \(\psi (x,0)=\psi _{0}(x)\) are added. The main result of the paper proves that \(\lim \left\| \psi _{i}(t)-\psi _{j}(t)\right\| _{L_{x}^{2}}=0\) if the coupling strength \(K\) is positive and if the maximal \(L^{2}\) distance between the initial wave functions is smaller than 1/2. The authors first recall some properties of the Schrödinger-Lohe model. They study the Cauchy problem for the free Schrödinger equation \(\partial _{t}\psi =i\Delta \psi \) with the initial condition \(\psi (x,0)=\psi _{0}(x)\) and the properties of the semigroup which may be associated to this problem. They explicit the link between the Schrödinger-Lohe model and the Kuramoto model. The proof of the main result is obtained through direct computations and using some Gronwall-type inequality. The paper ends with the proof of some \(H^{k}\) quantum synchronization for this model under further hypotheses on the data.