Phase space optimization of quantum representations: Direct-product basis sets

The quantitative phase space similarities between the uniformly mixed ensem bles of eigenstates, and the quasiclassical Thomas-Fermi distribution, are exploited in order to generate a nearly optimal basis representation for an arbitrary quantum system. An exact quantum optimization functional is prov ided, and the minimum of the corresponding quasiclassical functional is pro posed as an excellent approximation in the limit of large basis size. In pa rticular, we derive a stationarity condition for the quasiclassical solutio n under the constraint of strong separability. The corresponding quantum re sult is the phase space optimized direct-product basis-customized with resp ect to the Hamiltonian itself, as well as the maximum energy of interest. F or numerical implementations, an iterative, self-consistent-field-like algo rithm based on optimal separable basis theory is suggested, typically requi ring only a few reduced-dimensional integrals of the potential. Results are obtained for a coupled oscillator system, and also for the 2D Henon-Heiles system. In the latter case, a phase space optimized discrete variable repr esentation (DVR) is used to calculate energy eigenvalues. Errors are reduce d by several orders of magnitude, in comparison with an optimized sinc-func tion DVR of comparable size. (C) 1999 American Institute of Physics. [S0021 -9606(99)00235-4].