Commuting extensions and cubature formulae

Abstract

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A 1 , . . . , A d , related to the coordinate operators x 1 , . . . , x d , in R d . We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , . . . , A d , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

Matrices with normal defect one

“…In this section, we consider the commuting completion problem in the class of pairs of symmetric matrices. This is a special case of the problem raised in Degani et al [3] (see the first paragraph of Section 4) for N = 2. The authors of [3] presented an approach to n-dimensional cubature formulae where the cubature nodes are obtained by means of commuting completions of certain matrix tuples.…”

“…This is a special case of the problem raised in Degani et al [3] (see the first paragraph of Section 4) for N = 2. The authors of [3] presented an approach to n-dimensional cubature formulae where the cubature nodes are obtained by means of commuting completions of certain matrix tuples. While their commuting completion problem is stated in a certain subclass of tuples of symmetric matrices, some observations were also made for the problem in the whole class.…”

“…The problem of finding commuting completions of a N -tuple of n × n matrices was raised in [3], where a special emphasis was placed on symmetric completions of N -tuples of symmetric matrices. In [8], an inverse completion (A ext , B ext ) of a pair (A, B) was constructed.…”

Calculating Multidimensional Discrete Variable Representations from Cubature Formulas

“…Then, in a suitable basis of , the X̃ i are commuting extensions of the noncommuting coordinate matrices X i on . This actually solves a problem that was left open in refs and . It was not known there how to interpret as a function space the vector space on which commuting extensions act.…”

Efficient molecular quantum dynamics in coordinate and phase space using pruned bases

“…This approach is very successful for solving the time-independent Schrödinger equation, i. e. retrieving the eigenfunctions. It can be simply implemented by discarding functions that are located at points with high potential, 28,29 or by more advanced techniques like simultaneous diagonalization, [37][38][39] utilization of phasespace-structured basis functions [40][41][42][43] or sparse grids. [44][45][46] It has also been used for time-independent scattering simulations.…”