Three-dimensional Hyperbolic Spaces Research Articles

Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces

The description of the Paley–Wiener space for compactly supported smooth functions \(C^\infty _c(G)\) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for \(G=\textbf{SL}(2,\mathbb {R})^d\) (\(d\in \mathbb {N}\)) and \(G=\textbf{SL}(2,\mathbb {C})\). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

Path Integral for a Family of Classical Superintegrable Potentials

Simple integral formulations for a general potential depending on an arbitrary radial function in three-dimensional spherical and hyperbolic spaces of constant curvature are presented. The propagators are calculated in the usual 3D polar coordinates by using the Weyl-ordering prescription. By analyzing the radial function, the obtained expressions are compared with the path integrals for the Smorodinsky-Winternitz (SW) and generalized Kepler-Coulomb (GKC) systems. The results can be applied to obtain solvable path integral forms for interesting potentials.

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach. II

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, \documentclass[12pt]{minimal}\begin{document}$S_\kappa ^3$\end{document}Sκ3 (κ > 0) and \documentclass[12pt]{minimal}\begin{document}$H_k^3$\end{document}Hk3 (κ < 0), to the standard spherical waves in E3. The curvature κ is considered as a parameter and for any κ we show how the radial Schrödinger equation can be transformed into a κ-dependent Gauss hypergeometric equation that can be considered as a κ-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both κ-dependent and κ-independent), and are explicitly obtained.

Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, S3κ (κ > 0) and H3k (κ < 0), is studied. The curvature κ is considered as a parameter and then the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation that can be considered as a κ-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere S3κ (κ > 0) and the hyperbolic space H3k (κ < 0). A comparative study between the spherical and the hyperbolic quantum results is presented.

The Harmonic Oscillator on Three-Dimensional Spherical and Hyperbolic Spaces: Curvature Dependent Formalism and Quantization

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, \(S_{\kappa }^{3}\) (κ>0) and \(H_{k}^{3}\) (κ<0), is studied using geodesic spherical coordinates (r,θ,φ). The curvature κ is considered as a parameter and the results are formulated in explicit dependence of κ. The first part of the paper is concerned with the existence of Killing vectors, the existence of Noether symmetries and the properties of the Noether momenta. The second part is devoted to the transition from classical to quantum mechanics. The classical system is quantized by obtaining a κ-dependent invariant measure dμκ and expressing the Hamiltonian as a function of the Noether momenta.

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.

On zeros of the derivative of the three-dimensional Selberg zeta function

In this article, we study the distribution of zeros of the derivative of the Selberg zeta function for compact three-dimensional hyperbolic spaces. We obtain an asymptotic formula for the counting function of its zeros. This is a three-dimensional version of the celebrated work of Wenzhi Luo. We also deduce other asymptotic formulas relating to its zeros from the above formula.

Expository remarks on three-dimensional gravity and hyperbolic invariants

We consider complex invariants associated with compact real three-dimensional hyperbolic spaces. The contribution of the Chern–Simons invariants of irreducible U(n)-flat connections on hyperbolic fibered manifolds to the low-order expansion of the quantum gravitational path integral is analyzed.

Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces

We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells. Furthermore we compare these mean characteristics with those for Poisson-Voronoi tessellations in three-dimensional Euclidean spaces. It is shown that, as the absolute value of the curvature of hyperbolic spaces increases from zero to infinity, these mean characteristics increase monotonically from those for the Euclidean case to infinity.

Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces

We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells. Furthermore we compare these mean characteristics with those for Poisson-Voronoi tessellations in three-dimensional Euclidean spaces. It is shown that, as the absolute value of the curvature of hyperbolic spaces increases from zero to infinity, these mean characteristics increase monotonically from those for the Euclidean case to infinity.

Incidence theorems in spaces of constant curvature

Certain analogs of the classic theorems of Menelaus and Ceva are considered for a hyperbolic surface, a sphere, and for three-dimensional hyperbolic and spherical spaces.

Three-dimensional hyperbolic spaces

Three-dimensional hyperbolic spaces