Class of classically chaotic time-periodic Hamiltonians which lead to separable evolution operators.

Abstract

If V(r) is homogeneous in the position coordinates, then the Schr\"odinger equation with H(t)=T+\ensuremath{\alpha}(t)V(r) is separable in time if \ensuremath{\alpha}(t) is properly chosen. This fact makes the time evolution of such systems much easier to compute than it would otherwise be. Examples of such systems are the hydrogen atom with time-varying nuclear charge, the harmonic oscillator with a changing spring constant, and the infinite square well with moving walls. If \ensuremath{\alpha}(t) is made periodic by concatenating ``properly chosen'' segments, then a unitary operator U can be defined which evolves the system through one period. If the wave function is expressed as an N-term truncated expansion of eigenfunctions of H(t), then U is an N\ifmmode\times\else\texttimes\fi{}N matrix which can be computed straightforwardly. The properties of U are studied by examining the statistical properties of its eigenvectors and eigenvalues, whose phases are the quasienergies. Numerical examples are given for the Fermi-Ulam cosmic accelerator (FUCA), which is a particle in a one-dimensional box with a moving wall, with various values for relevant parameters, which can lead to Poisson, Gaussian orthogonal ensemble, Gaussian unitary ensemble, or other quasienergy statistics, corresponding to inte- grable, chaotic, and non-time-reversible chaotic situations. Poincar\'e sections for the classical FUCA are also displayed. They show regimes of regular and of chaotic motion, whose correspondence with the statistics of U is sought with limited success.