Quantum quasi-Lie systems: properties and applications

Abstract

A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with $t$-dependent Schrödinger equations determined by a particular class of $t$-dependent Hamiltonian operators, the quantum Lie systems, and other differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with $t$-dependent Schrödinger equations associated with the here called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum $t$-dependent Smorodinsky--Winternitz oscillators.

References in corpus (5)

• Superposition rules, Lie theorem and partial differential equations
• Geometry of quantum systems: density states and entanglement
• The time-dependent quantum harmonic oscillator revisited: Applications to Quantum Field Theory
• Queer Poisson brackets
• Multisymplectic structures and invariant tensors for Lie systems