We study the asymptotic emergent dynamics and the continuum limit for the Schrödinger-Lohe (SL) model and semi-discrete SL model. For the SL model, emergent dynamics has been mostly studied for systems with identical potentials in literature. In this paper, we further extend emergent dynamics and stability estimate for the SL model with nonidentical potentials. To achieve this, we use two-point correlation functions defined as an inner product between wave functions. For the semi-discrete SL model, we provide a global unique solvability and a sufficient framework for the smooth transition from the semi-discrete SL model to the SL model in any finite-time interval, as the mesh size tends to zero. Our convergence estimate depends on the uniform-in- h h Strichartz estimate and the uniform-stability of the SL models with respect to initial data.

We study emergent behaviors of the Lohe hermitian sphere(LHS) model with a time-delay for a homogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit hermitian sphere in C d with d ≥ 2, Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model [23]. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on R 2d . In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

<p style='text-indent:20px;'>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.</p>