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Symmetry Reduction of Fourier Kernels

✍ Scribed by J.H Samson; G.A Evans

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
253 KB
Volume
142
Category
Article
ISSN
0021-9991

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✦ Synopsis

Fourier transforms of functions of several variables invariant under certain symmetry groups are studied, with particular reference to functions f (x 1 β€’ β€’ β€’ x N ) of N three-component vectors invariant under rigid rotations. Here we use symmetry to enhance the efficiency of evaluation of the integrals. The Fourier transform can be written as an integral

exp(i A zz (2u -1)) du, a function of the eigenvalues of the dyadic A = ͚ N i=1 k i x i . For N = 1 the familiar Hankel transform is recovered. For N = 2 the kernel reduces to a single integral of elementary functions, equal to the local spin-flip propagator in a one-dimensional tight-binding antiferromagnet. A variety of forms is given, and useful asymptotic forms are found in various limits. Recent numerical methods for the evaluation of irregular oscillatory integrals are applied to the kernel in the N = 2 case.

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