Notes on the Riccati operator equation in open quantum systems

TL;DR

This paper addresses the Riccati operator equation in open quantum systems, solving a recent problem related to antilinear solutions and highlighting potential pitfalls in simplifying the equation.

Contribution

It provides a solution to a recent open problem about antilinear Riccati solutions and demonstrates that simplifying the Riccati equation can lead to non-equivalent equations.

Findings

Solved the open problem on antilinear Riccati solutions

Showed that simplifications can lead to non-equivalent equations

Clarified the structure of Riccati equations in quantum systems

Abstract

A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.