@article {1468,
	title = {Spectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: Comparison with sinc methods},
	journal = {Journal of Chemical Physics},
	volume = {125},
	number = {19},
	year = {2006},
	note = {ISI Document Delivery No.: 107OVTimes Cited: 3Cited Reference Count: 76Lo, Joseph Shizgal, Bernie D.},
	month = {Nov},
	pages = {17},
	type = {Article},
	abstract = {Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I-2 with a Morse potential is also considered. (c) 2006 American Institute of Physics.},
	keywords = {BOLTZMANN COLLISION OPERATOR, DISCRETE-VARIABLE REPRESENTATIONS, EIGENVALUE PROBLEMS, ELECTRON THERMALIZATION, LANCZOS METHOD, MATRIX-ELEMENTS, NONCLASSICAL ORTHOGONAL POLYNOMIALS, ORDINATE METHOD, SINGULAR CONVOLUTION, SUPERSYMMETRIC QUANTUM-MECHANICS},
	isbn = {0021-9606},
	url = {<Go to ISI>://000242181800010},
	author = {Lo, J. and Shizgal, B. D.}
}
@article {7109,
	title = {COMPARISON OF WKB (WENTZEL-KRAMERS-BRILLOUIN) AND SWKB SOLUTIONS OF FOKKER-PLANCK EQUATIONS WITH EXACT RESULTS - APPLICATION TO ELECTRON THERMALIZATION},
	journal = {Canadian Journal of Physics},
	volume = {69},
	number = {6},
	year = {1991},
	note = {ISI Document Delivery No.: FZ294Times Cited: 5Cited Reference Count: 41},
	month = {Jun},
	pages = {712-719},
	type = {Article},
	abstract = {A comparison of WKB (Wentzel-Kramers-Brillouin) and SWKB eigenfunctions of the Schrodinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker-Planck eigenvalue problem to a Schrodinger equation. Linear Fokker-Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. The specific Fokker-Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.},
	keywords = {ACTIVATED, APPROXIMATION, DISCRETE-ORDINATE METHOD, EIGENVALUES, INVARIANCE, ISOMERIZATION, NUCLEATION, RARE-GAS MODERATORS, RATE-PROCESSES, SHAPE, SOLVABLE POTENTIALS, SUPERSYMMETRIC QUANTUM-MECHANICS},
	isbn = {0008-4204},
	url = {<Go to ISI>://A1991FZ29400011},
	author = {Shizgal, B. and Demeio, L.}
}
