Statistics – Computation
Scientific paper
Feb 1989
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989a%26a...210...18s&link_type=abstract
Astronomy and Astrophysics (ISSN 0004-6361), vol. 210, no. 1-2, Feb. 1989, p. 18-24. Research supported by the University of Shi
Statistics
Computation
10
Computational Astrophysics, Galaxies, Liouville Equations, Stellar Systems, Angular Momentum, Distribution Functions, Eigenvalues, Hilbert Space, Time Dependence
Scientific paper
Time-dependent solutions of Liouville's equations are constructed. From a review of existing literature on such solutions, it is concluded that the eigenfunctions of Liouville's operator are complex functions of phase coordinates in a complex Hilbert space. The latter in turn is the direct product of two Hilbert spaces, one accommodating functions of configuration coordinates and the other those of the momentum coordinates. A pair of parity operators Q and P in configuration and momentum space, respectively, is introduced. It is shown that QP commutes with the Liouville operators and classifies the latter's eigenfunctions on the basis of their parities. A pair of angular momentum operators L and K in configuration and momentum spaces, respectively, is introduced, and it is shown the J = L + K commutes with the Liouville operator and QP. The simultaneous eigenfunctions of the mutually commuting operators L, J-squared, J(z), and QP are constructed.
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