Quantum quasi-Lie systems: properties and applications

• URL: http://arxiv.org/abs/2204.00954v1
• Date: Sat, 2 Apr 2022 23:25:26 GMT
• Title: Quantum quasi-Lie systems: properties and applications
• Authors: J.F. Cari\~nena, J. de Lucas, and C. Sard\'on
• Abstract summary: A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values. This work extends quasi-Lie schemes and quantum Lie systems to cope with $t$-dependent Schr"odinger equations.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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