Lame functions

Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

So how is the spherical pendulum quantised The answer is that its motion is generally chaotic, except for discrete values of the initial projection speed, for which it is periodic. The precise details of this phenomenon are difficult to get a handle on, because, although the vertical motion is always described by periodic elliptic functions, the horizontal motion is described in terms of Lame functions, which are very difficult to study and for which periodicity is difficult to diagnose. 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

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. 

I even and Vl - w, /w, Vl + w, w - xo ) for I odd define the eight types of Lame functions in their different species, as in Section 2.1, upon multiplication by their respective polynomials P(w). 

The numerical results for the eigenenergies and eigenfunctions evaluated in Refs. [5] and [6] for molecules with different asymmetry distributions and states are accurate and consistent. The zeros of the individual Lame functions can be determined with high accuracy, and are illustrated in Figure 1 in Ref. [6]. They allow writing the Lame polynomial in product forms presented in Sections 2.1 and 2.2. They are also the key to implement the boundary condition for the rotations of molecules confined by elliptical cones as discussed in Section 3. 

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... 

We also include the differential equations for fhe remaining factor in the Lame functions after removing the singularity factors... 

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. 

For the mathematically inclined Volkmer s Chapter 29 on Lame functions in the NIST Handbook of Mathematical Functions [50], and the article "A new basis for the representations of the rotational group. Lame and Heun polynomials" [51] are parts of the toolbox. 

X.L. Ka, Lame Functions. Advanced Quantum Mechanics, Higher Education, Beijing, New York, NY, USA, 2001, p. 135. 

Correspondingly, the spheroconal harmonics to be used in the eigenfunctions of the hydrogen atom, Equation (48), are the products of Lame functions in the respective variables, with common values of l and... 

The products AB agree with the transformation Equation (45) and their parity properties under reflection in the respective cartesian coordinate planes.The sub-indexes in Equation (57) count the number of nodes of the respective Lame functions, and their sum is such that... 

---