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

Abstract

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 that 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 that we evaluate using a recent method of integrating the quantum Liouville-Bloch equations [A. R. P. Rau, Phys. Rev. Lett. 81, 4785 (1990)]. 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 $\mathrm{su}(1,1)$ algebra and that of simple dynamical systems.