Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator

Details Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator

2000-12-22

arxiv.org/abs/nlin/0012046v1

Phys. Rev. A, vol. 63, article # 042109 (2001)

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages, 4 figures, Accepted in PRA

Scientific paper

10.1103/PhysRevA.63.042109

The classical and the quantal problem of a particle interacting in one-dimension with an external time-dependent quadratic potential and a constant inverse square potential is studied from the Lie-algebraic point of view. The integrability of this system is established by evaluating the exact invariant closely related to the Lewis and Riesenfeld invariant for the time-dependent harmonic oscillator. We study extensively the special and interesting case of a kicked quadratic potential from which we derive a new integrable, nonlinear, area preserving, two-dimensional map which may, for instance, be used in numerical algorithms that integrate the Calogero-Sutherland-Moser Hamiltonian. The dynamics, both classical and quantal, is studied via the time-evolution operator which we evaluate using a recent method of integrating the quantum Liouville-Bloch equations \cite{rau}. The results show the exact one-to-one correspondence between the classical and the quantal dynamics. Our analysis also sheds light on the connection between properties of the SU(1,1) algebra and that of simple dynamical systems.

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