Lame functions eigenvalue

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

Table 4.6 Second eigenvalue h]. and matching h of the respective Lame functions as functions of angular momentum values k for the successive families of elliptical cones > > 0... 

Notice that the coefficient + 1) in the original equation becomes — 1) + 2), ( — 2)( + 3), and ( — 3) ( + 4) for the successive species being the same as Eqs. (52-56), while the eigenvalues hf are also shifted depending on the coefficients involved in the second derivatives of fA(Xi)- These changes are reminiscent of the familiar ones for the ordinary and associated Legendre polynomials, and their connections with the actions of ladder operafors. We are exploring the possibilities for the Lame functions themselves and their connections with Section 4.2.2. 

Table II in Ref. [6] illustrates the eigenvalues h for the cases of = 4 and 5 for the respective species and types of the Lam6 polynomials for molecules with the different asymmetry distributions. Figure 1 in Ref. [6] shows the variations of the Lame polynomials A% A% A (three of each), and A " (two) as functions of their argument and of the asymmetry distribution.

The Lame quasi-periodic functions are common eigenfunctions of fhe operators U and H as discussed in Sections 2.2 and 2.5. In Refs. [1] and [8], we chose the notation of G for the latter in order to emphasize that it represents the geometry of the spheroconal elliptical cone confinement, in contrast with its dynamical character for the rotations of asymmetric molecules. The eigenvalues and /i are numerically found to satisfy the relationships... 

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