Lie Algebra Contractions and Interbasis Expansions on Two-Dimensional Hyperboloid IIA. Subgroup Basis

TL;DR

The paper analyzes how three subgroup-type eigenfunctions of the Laplace-Beltrami operator on the two-sheeted hyperboloid H2+ contract to eigenfunctions on the Euclidean plane E2 as the contraction parameter ε = 1/R → 0. It constructs horocyclic, pseudo-spherical, and equidistant bases, derives explicit interbasis expansions among them, and shows that the expansion coefficients contract to familiar flat-space objects such as Bessel and plane-wave representations. Interbasis kernels are expressed in terms of Gamma functions, Wilson-Racah polynomials, and Laguerre sums, with rigorous orthogonality and completeness preserved under contraction. The results provide a unified view of how curved-space eigenfunctions and their interbasis relations reduce to standard Euclidean structures, enabling integral representations and facilitating applications in spectral theory and mathematical physics.

Abstract

Three subgroup type eigenfunctions of the Laplace-Beltrami operator on a two-dimensional two-sheeted hyperboloid are considered and all interbasis expansions between them are calculated. It is shown how the coefficients determining the expansions and the expansions themselves between subgroup basis contract from the hyperboloid to the Euclidean plane.

Figures (23)

• Figure 1: Wave function $\sqrt{\tilde{y}} K_{i\rho}(|s|\tilde{y})$ for $\rho = s = 4$.
• Figure 2: Graphics of potential $V_m^S(\tau)$ for $m = 0$ (red line), $m = 1$ (blue line) and $m = 2$ (green line).
• Figure 3: Garphics of function $N_{\rho m} P^{|m|}_{i\rho-1/2}(\cosh\tau)$ for $\rho = 1$, $R = 1$, $m = 0$ (red line) and $m = 2$ (green line).
• Figure 4: Graphics of potential $V^{EQ}_\nu(\tau_1)$ for $\nu = 0$ (blue line), $\nu = 1/2$ (green line) and $\nu = 1$ (red line).
• Figure 5: Graphics of wave functions $\psi_{\rho\nu}^{(+)}$ (solid) and $\psi_{\rho\nu}^{(-)}$ (dashed).