where d 3 3)2 ( L x - -- i3x j3x j i i>j Thus the Gegenbauer polynomials play a role in the theory of hyper spherical harmonics which is analogous to the role played by Legendre polynomials in the familiar theory of 3-dimensional spherical harmonics; and when d = 3, the Gegenbauer polynomials reduce to Legendre polynomials. The familiar sum rule, in 'lrlhich a sum of spherical harmonics is expressed as a Legendre polynomial, also has a d-dimensional generalization, in which a sum of hyper spherical harmonics is expressed as a Gegenbauer polynomial (equation (3-27»: The hyper spherical harmonics which appear in this sum rule are eigenfunctions of the generalized angular monentum 2 operator A , chosen in such a way as to fulfil the orthonormality relation: VIe are all familiar with the fact that a plane wave can be expanded in terms of spherical Bessel functions and either Legendre polynomials or spherical harmonics in a 3-dimensional space. Similarly, one finds that a d-dimensional plane wave can be expanded in terms of HYPERSPHERICAL HARMONICS xii "hyperspherical Bessel functions" and either Gegenbauer polynomials or else hyperspherical harmonics (equations ( 4 - 27) and ( 4 - 30) ) : 00 ik·x e = (d-4)!!A~oiA(d+2A-2)j~(kr)C~(~k'~) 00 (d-2)!!I(0) 2: iAj~(kr) 2:Y~ (["2k)Y (["2) A A=O ). l). l)J where I(O) is the total solid angle. This expansion of a d-dimensional plane wave is useful when we wish to calculate Fourier transforms in a d-dimensional space.
`Since the book is clearly written, it will be useful to students and research workers in mathematics, nuclear physics, atomic physics and quantum chemistry.'
G. van den Berghe in Mathematical Reviews
Harmonic polynomials.- Generalized angular momentum.- Gegenbauer polynomials.- Fourier transforms in d dimensions.- Fock's treatment of hydrogenlike atoms and its generalization.- Many-dimensional hydrogenlike wave functions in direct space.- Solutions to the reciprocal-space Schrodinger equation for the many-center Coulomb problem.- Matrix representations of many-particle Hamiltonians in hyper spherical coordinates.- Iteration of integral forms of the Schrodinger equation.- Symmetry-adapted hyperspherical harmonics.- The adiabatic approximation.- Appendix A: Angular integrals in a 6-dimensional space.- Appendix B: Matrix elements of the total orbital angular momentum operator.- Appendix C: Evaluation of the transformation matrix U.- Appendix D: Expansion of a function about another center.- References.