@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 {3811,
	title = {The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions},
	journal = {Journal of Chemical Physics},
	volume = {104},
	number = {11},
	year = {1996},
	note = {ISI Document Delivery No.: TY728Times Cited: 34Cited Reference Count: 66},
	month = {Mar},
	pages = {4137-4150},
	type = {Article},
	abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker-Planck equations. The Fokker-Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics, For this class of potentials, the groundstate wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker-Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrodinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O-2 and Ar-Xe with realistic pair potentials. For O-2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrodinger equation with the Henon-Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrodinger equation is very rapid with this approach. (C) 1996 American Institute of Physics.},
	keywords = {BOUND-STATES, COEFFICIENTS, DISCRETE-ORDINATE METHOD, EIGENVALUE PROBLEMS, FOKKER-PLANCK EQUATION, ORTHOGONAL POLYNOMIALS, POTENTIALS, QUANTUM-MECHANICS, RECURRENCE, ROVIBRATIONAL STATES},
	isbn = {0021-9606},
	url = {<Go to ISI>://A1996TY72800030},
	author = {Shizgal, B. D. and Chen, H.}
}
