A general framework for discrete variable representation basis sets

Abstract

A framework for discrete variable representation ͑DVR͒ basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected ␦-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay ͓J. Chem. Phys. 105, 6940 ͑1996͔͒ gives exponential convergence.

Effective many-body parameters for atoms in nonseparable Gaussian optical potentials

“…However, the components of V (x) x| n which lie outside of S K are typically O(1/N ) and this error is concentrated in the basis elements near the boundary n N . For wave functions well converged with a basis size N , the weight on the basis functions with n near N is exponentially small, and hence we recover exponential accuracy [21].…”

“…Hence, the loss in sparsity of the DVR compared to finite-order finite-differencing methods is more than compensated by the increase in accuracy. Generally speaking, exponential convergence comes only from using an analytic basis set for expansion [21], and so methods based on nonanalytic wave functions, such as Haar wavelets, B splines, and finite differencing, will not display exponential convergence.…”

“…We define a DVR as follows [21]. Given a domain D ⊆ R and a Hilbert space H of functions on D, we would like to find a subspace S K of H with refinement parameter K which captures the part of H spanned by low-energy eigenstates of our potential.…”

Generalized Non-Relativistic Supersymmetric Quantum Mechanics

“…For W(x)=x 3 , and thus V(x)=x 6 − 3x 2 , we carried out a variational calculation using the system-specific coherent states defined above and compared the accuracy in the approximation of the first three excited state energy eigenvalues with that achieved using the standard harmonic oscillator basis and the harmonic oscillator coherent states. To evaluate the accuracy of each method, we compare the results with a Chebyshev polynomial DVR (Discrete Variable Representation) calcuation using 1000 points Littlejohn (2002). The number of decimal places reported in Tables 7-10 correspond to the number of decimal places of agreement with the DVR plus an additional significant figure which is either rounded up or down.…”

Commuting extensions and cubature formulae

“…This paper was born out of an attempt to extend the DVR method to higher dimensions; other, recent progress on this subject has been made by Dawes and Carrington [9]. Another interesting perspective on DVR can be found in the paper [10].…”