Symmetric Functions and Bn -invariant Spherical Harmonics
نویسنده
چکیده
The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type weight function on the sphere. Their analysis is carried out by means of differential-difference operators. The symmetries of this system involve the Weyl group of type B, generated by permutations and changes of sign of the coordinates. A new basis for symmetric functions as well as an explicit transition matrix to the monomial basis is constructed. This basis leads to a basis for invariant spherical harmonics. The determinant of the Gram matrix for the basis in the natural inner product over the sphere is evaluated, and there is a formula for the evaluation of the basis elements at (1,1,...,1). When the underlying parameter is specialized to zero, the basis consists of ordinary spherical harmonics with cube group symmetry, as used for wave functions of electrons in crystals. The harmonic oscillator can also be considered as a degenerate interaction-free spin Calogero model.
منابع مشابه
Integral Properties of Zonal Spherical Functions, Hypergeometric Functions and Invariant
Some integral properties of zonal spherical functions, hypergeometric functions and invariant polynomials are studied for real normed division algebras.
متن کاملAn efficient routine for computing symmetric real spherical harmonics for high orders of expansion
Author(s) of this paper may load this reprint on their own web site provided that this cover page is retained. Republication of this article or its storage in electronic databases or the like is not permitted without prior permission in writing from the IUCr. A numerically efficient method of constructing symmetric real spherical harmonics is presented. Symmetric spherical harmonics are real sp...
متن کاملTraceless Symmetric Tensor Approach to Legendre Polynomials and Spherical Harmonics
In these notes I will describe the separation of variable technique for solving Laplace’s equation, using spherical polar coordinates. The solutions will involve Legendre polynomials for cases with azimuthal symmetry, and more generally they will involve spherical harmonics. I will construct these solutions using traceless symmetric tensors, but in Lecture Notes 8 I describe how the solutions i...
متن کاملDetermination of Fiber Direction in High Angular Resolution Diffusion Images using Spherical Harmonics Functions and Wiener Filter
Diffusion tensor imaging (DTI) MRI is a noninvasive imaging method of the cerebral tissues whose fibers directions are not evaluated correctly in the regions of the crossing fibers. For the same reason the high angular resolution diffusion images (HARDI) are used for estimation of the fiber direction in each voxel. One of the main methods to specify the direction of fibers is usage of the spher...
متن کاملFast Spin ±2 Spherical Harmonics Transforms
An exact fast algorithm is developed for the direct spin-weighted spherical harmonics transforms of bandlimited spin ±2 functions on the sphere. First, we define spin functions on the sphere and their decomposition in an orthonormal basis of spin-weighted spherical harmonics. Second, we discuss the a priori O(L4) asymptotic complexity of the spin ±2 spherical harmonics transforms, where 2L stan...
متن کامل