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

摘要

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