On the complete aggregation of the Wigner-Lohe model for identical potentials

We study the collective behaviors of the Wigner-Lohe (WL) model for quantum synchronization in phase space which corresponds to the phase description of the Schrödinger-Lohe (SL) model for quantum synchronization, and it can be formally derived from the SL model via the generalized Wigner transform. For this proposed model, we show that the WL model exhibits asymptotic aggregation estimates so that all the elements in the generalized Wigner distribution matrix tend to a common one. On the other hand, for the global unique solvability, we employ the fixed point argument together with the classical semigroup theory to derive the global unique solvability of mild and classical solutions depending on the regularity of initial data.