Quantum dynamical semigroups for diffusion models with Hartree interaction

TL;DR

This paper establishes the existence and uniqueness of global solutions for a class of nonlinear quantum dynamical equations with mean-field interactions, including the Quantum Fokker-Planck-Poisson model, advancing the mathematical understanding of dissipative quantum systems.

Contribution

It proves the existence of a nonlinear conservative quantum dynamical semigroup for Lindblad-type equations with Hartree interaction, addressing unbounded generators and nonlinearity.

Findings

Proved global-in-time existence and uniqueness of solutions

Established mass preservation in the quantum evolution

Demonstrated the well-posedness of a class of nonlinear quantum models

Abstract

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.