Tomaschitz, R. (1989). On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature, Physica D 34, 42-89, DOI: 10.1016/0167-2789(89)90228-5

 

 

Abstract (ScienceDirect, CDS, SAO/NASA ADS, Zbl 0702.58079)

We consider geodesic motion on three-dimensional Riemannian manifolds of constant negative curvature, topologically equivalent to S × ]0, 1[, S a compact surface of genus two. To those trajectories which are bounded and recurrent in both directions of the time evolution t → + ∞, t → − ∞, a fractal limit set is associated whose Hausdorff dimension is intimately connected with the quantum mechanical energy ground state, determined by the Schrödinger operator on the manifold.

  We give a rather detailed and pictorial description of the hyperbolic spaces we have in mind, discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and Hausdorff dimension, and give finally some examples of ground state calculations in terms of Hausdorff dimensions of limit sets of classical trajectories.

 

 

 

Zbl 0702.58079
Tomaschitz, R.
On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature
[J] Physica D 34, No.1-2, 42-89 (1989). ISSN 0167-2789

MSC 2000:

*58J60 Relations with special manifold structures
81Q05 Closed and approximate solutions to quantum-mechanical equations
58Z05 Appl. of global analysis to physics
37A99 Ergodic theory
53C20 Riemannian manifolds (global)

Keywords: Laplace-Beltrami operator; Hausdorff dimension; energy ground state; geodesic motion

 

Title:		On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature
Authors:		Tomaschitz, R.
Affiliation:		AA(Sechsschimmelg, 1/21-22, A-1090 Vienna, Austria)
Publication:		Physica D: Nonlinear Phenomena, Volume 34, Issue 1-2, p. 42-89.
Publication Date:		01/1989
Origin:		ELSEVIER
DOI:		10.1016/0167-2789(89)90228-5
Bibliographic Code:		1989PhyD...34...42T

 

 

Plates I-X: The description of the plates is given in section 8.

 

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description: Roman Tomaschitz (1989) On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature, Physica D 34, 42.

 

Keywords: Laplace–Beltrami operator and Schrödinger equation on hyperbolic 3-manifolds, multiply connected topology, spectral decomposition of automorphic wave fields, scalar point-pair invariants, Poisson kernel, Green’s function, Poincaré series, Möbius transformations in the Poincaré half-space, universal covering space, polyhedral tessellation of hyperbolic space, fundamental polyhedra and fractal limit sets of quasi-Fuchsian covering groups, Jordan curves, fibered hyperbolic 3-manifolds, compact Riemann surfaces, fundamental domains of discrete subgroups of the Lorentz group, Hausdorff dimension of limit sets, covering projection, quantum mechanical ground-state energy and recurrent geodesics, deformation spaces of open hyperbolic 3-manifolds

 

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