Quantum quasi-Lie systems: properties and applications

TL;DR

This paper introduces quantum quasi-Lie systems, extending existing Lie system frameworks to analyze $t$-dependent Schrödinger equations, with applications to quantum oscillators and fluid models.

Contribution

It extends quasi-Lie schemes and quantum Lie systems to include quantum quasi-Lie systems for $t$-dependent Schrödinger equations, broadening analytical tools.

Findings

Developed the theory of quantum quasi-Lie systems.

Applied the framework to quantum nonlinear oscillators.

Analyzed quantum fluid dynamics in trapping potentials.

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 Schr\"odinger 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 Schr\"odinger 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.