Spheroconal harmonics

The identification of the solutions of Eq. (13) as Lam6 spheroconal harmonic functions in the respective coordinates allows writing them as superpositions of the familiar spherical harmonics ... 

Niven established the connections between the ellipsoidal harmonics, expressed in cartesian coordinates, and the spheroconal harmonics, expressed in spheroconal coordinates, in the respective factors of Eq. (18), by requiring that the eigenfunctions h satisfy the Laplace equation [18]. The application of the Laplace operator on the eigenfunctions with the condition of vanishing leads to the zeros 0, of the respective polynomials, which are real and different in their respective domains ccartesian coordinates leads to the corresponding condition for its being harmonic... 

The expansion of Eq. (62) can be inverted to express the spherical harmonics in terms of spheroconal harmonics, whose completeness in turn leads to the addition theorem in the form... 

The construction of the Lam6 spheroconal harmonic polynomials involves Eq. (43) with matching parameters Eq. (44), matching species, and matching excitations of the respective elliptical cone coordinate degrees of freedom. The matching of species and kinds are fhe following ... 

The equal sfanding of spherical harmonics and spheroconal harmonics is manifested in this case. The rotations of the hydrogen atom are familiarly described fhrough eigenfunctions of and U, buf fhey also admif eigenfunctions of H and R. The same holds for any cenfral pofenfial quan-fum system. Some of fhe consequences are illusfrafed in fhe following section. 

The complete Lame spheroconal harmonic quasi-periodic functions, satisfying the respective boundary conditions in Eq. (91), involve matching the corresponding factors of Table 4.4 and their counterparts. For the boundary... 

Table 4.4 Matching species of Lam6 functions whose products are Lame spheroconal harmonic functions with defined parities...

ON DEVELOPING THE THEORY OF ANGULAR MOMENTUM IN BASES OF LAhAi SPHEROCONAL HARMONICS... 

In Section 4.1, the identification of complefe radial and angular momentum raising and lowering operators for fhe familiar spherical harmonics is presented in our own version, as a point of reference for some extensions in Section 4.2 for fhe spheroconal harmonics. The resulfs in Section 4.1 have been known in the literature [44, 46], but here the interest is in their adaptation and extension to eigenfunctions of P and Lf, and and H. ... 

The method and results of this section also illuminate some of our works on rotations of asymmetric molecules, the hydrogen atom and any central potential in the direction of developing the theory of angular momentum in the Lam6 spheroconal harmonic bases, as discussed in the following sections. 

Angular momentum transformations of Lam6 spheroconal harmonic polynomials... 

The familiar theory of angular momentum, based on spherical harmonics, eigenfunctions of and operators of angular momentum, also uses the ladder operators L , which for a given value of i connect all the 2 - -1 successive states m) with m = —I, -I + 1,..., —1,0,1,..., -1,1. Section 4.2.1 illustrates the counterpart for the spheroconal harmonics or the spherical harmonics with well-defined parities under the application of the operators... 

Ly, L. Correspondingly, Section 4.2.2 illustrates for the spheroconal harmonics what has been rigorously proven for spherical harmonics in Section 4.1. Then, Section 4.2.3 identifies some spheroconal fools to further explore the development of the theory of angular momenfum. 

They also correspond to spheroconal harmonics of fhe species and fypes... 

The topics of this chapter, listed in the Contents and outlined in the Introduction, discuss the rotations of asymmetric molecules and the hydrogen atom in their natural free configurations as reviewed in Section 2 changes in the properties of the same systems in configurations of confinement by elliptical cones are reported, including new results for asymmetric molecules, in Section 3 and some advances in developing the theory of angular momentum in spheroconal harmonic bases, as well as some possible routes under exploration are presented in Section 4. 

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

R. Mendez-Fragoso, E. Ley-Koo, Lam6 spheroconal harmonics in atoms and molecules. Int. J. Quant. Chem. 110 (2010) 2765. 

Ghapter 4 presents a review of fhe exacf formulafions and evaluafions for the rotations of free asymmefric molecules and for treafment of fhe sysfems when confined by boundary condifions including elliptical cones. In addition, some tools and advances in development in the theory of angular momentum in bases of spheroconal harmonics are discussed. 

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