Related papers: Quantum quasi-Lie systems: properties and applicat…
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to…
A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie…
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…
The recently developed theory of quasi-Lie schemes is studied and applied to investigate several equations of Emden type and a scheme to deal with them and some of their generalisations is given. As a first result we obtain t-dependent…
A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie…
We prove that $t$-dependent Schr\"odinger equations on finite-dimensional Hilbert spaces determined by $t$-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot--Guldberg Lie algebra of K\"ahler…
We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of…
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…
A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the…
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they…
Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum…
A stochastic Lie system on a manifold $M$ is a stochastic differential equation whose dynamics is described by a linear combination with functions depending on $\mathbb{R}^\ell$-valued semi-martigales of vector fields on $M$ spanning a…
Lie algebroids provide a natural medium to discuss classical systems, however, quantum systems have not been considered. In aim of this paper is to attempt to rectify this situation. Lie algebroids are reviewed and their use in classical…
The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of…
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used.…
A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{\alpha=1}^rg_\alpha X_\alpha$, where $X_1,\ldots,X_r$ are vector fields on…
Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup…