Scattering Theory for Leaky Tori

Analytic properties of the scattering amplitude for Coulomb potential on the background of the space of constant negative curvature are studied. Special attention is given to the comparison of the Regge trajectories for curved and flat spaces. We show that there exist considerably differences in the behavior of the Regge trajectories in these spaces.

The present work is motivated by the problem of Bayesian inference for Gaussian distributions in symmetric Hadamard spaces (that is, Hadamard manifolds which are also symmetric spaces). To investigate this problem, it introduces new tools for Markov Chain Monte Carlo, and convex optimisation: (1) it provides easy-to-verify sufficient conditions for the geometric ergodicity of an isotropic Metropolis-Hastings Markov chain, in a symmetric Hadamard space, (2) it shows how the Riemannian gradient descent method can achieve an exponential rate of convergence, when applied to a strongly convex function, on a Hadamard manifold. Using these tools, two Bayesian estimators, maximum-a-posteriori and minimum-mean-squares, are compared. When the underlying Hadamard manifold is a space of constant negative curvature, they are found to be surprisingly close to each other. This leads to an open problem: are these two estimators, in fact, equal (assuming constant negative curvature)?KeywordsHadamard manifoldGaussian distributionBayesian inferenceMCMCConvex optimisation

The metrics of visual and haptic space based on parallelity judgements

A noncompact, nonintegrable billiard of importance in cosmology---the infinite equilateral triangle on a space of constant negative curvature---gives rise to a natural noncompact integrable approximation, which is studied here quantum mechanically. The Weyl formula for the averaged spectral staircase contains a nonstandard logarithmic term due to the noncompactness of the billiard. For two symmetry classes we determine numerically the spectral staircase, the level-spacing distribution p(s), and the averaged ${\mathrm{\ensuremath{\Delta}}}_{3}$ statistic, and compare with analytical expressions that follow from Poisson statistics and semiclassical theory. The Poissonian result for the correlation function of ${\mathrm{\ensuremath{\Delta}}}_{3}$ is derived in closed form and also compared with the data.

Visual space is, according to Luneburg, a non-Euclidean hyperbolic space of constant negative curvature. Two personal constants enter into the geometrical description of binocular subjective space. One is σ indicating the degree of depth perception characteristic to the observer. The second is K, the curvature; a value which denotes by how much the visual space of an individual departs from Euclidean space. The constants σ and K were determined experimentally for 30 untrained observers by Luneburg's 3- and 4-point tests. All values found for K were negative and support the hypothesis that the geometry of binocular space is hyperbolic. Some discrepancies between the theory and findings were observed, while the sensitivity of the experiments to the lack of training and even a slight measure of aniseikonia is evident.

The two-dimensional one-component logarithmic Coulomb gas is mapped onto a non-hermitian fermionic field theory. At $\beta=2$, the field theory is free. Correlation functions are calculated and a perturbation theory is discussed for extending to other $\beta$. A phase transition is found at the mean-field level at large $\beta$. Some results are extended to spaces of constant negative curvature.

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

It is shown that no non-flat empty space-time can be embedded locally and isometrically in a five-dimensional space of nonzero constant curvature.

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses move on a circle of not necessarily constant size has to be either a regular cocircular homographic orbit in flat space, or a regular polygonal rotopulsator in curved space, under the constraint that the minimal distance between point masses attains its minimum in finite time. Additionally, we prove that the same holds true if we add an extra mass at the center of that circle and find an explicit formula for the mass of each point particle in terms of the radius of the circle. Finally, we prove that for each order of the masses there is at most one cocircular homographic orbit for the case that the masses need not be constant.

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

The simplest chaotic Hamiltonian systems arise in geometry: a point particle moving freely on a 2 or 3 dimensional manifold of constant negative curvature. There is tremendous variety representing both bound state and scattering situations, quite in contrast to the non-chaotic motion on surfaces of constant positive curvature. The classical mechanics has been studied for almost a century and reduces to an enumeration and description of geodesic lines which in turn are closely connected with the topology of the manifold. Physics enters only in a trivial manner since everything can be scaled with the (kinetic) energy of the particle. In the quantum mechanics of these motions, however, the energy is the essential parameter, and one can ask how the two mechanics are related to each other. Quite surprisingly, Feynman's path integral gives the same answer as its classical approximation, just as in the harmonic oscillator. In 3 dimensions, this result holds for the propagator itself, while in 2 dimensions it is true only for the trace and yields Selberg's trace formula. A few striking examples of constant negative curvature spaces will be shown. Thus, quantum chaos can be understood in terms of classical chaos in spite of their qualitatively different features: fractal for the latter, smooth for the former.

For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R − r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient R/r. 1. Preliminaries and the main results In [1] A. Borisenko and V. Miquel proved that a closed hypersurface with normal curvatures kn satisfying the inequality kn > 1 in the Lobachevsky space H(−1) can be put into a spherical shell between two concentric spheres of radii R and r such that the width R−r of the shell satis esR−r 6 ln 2. A similar estimate holds in Hadamard manifolds (see [2]). In [3] these results were extended for Riemannian manifolds of constant-signed sectional curvatures and hypersurfaces with normal curvatures bounded below. In the present paper we re ne some results from [3]. For this purpose we consider hypersurfaces with normal curvatures at any point and in any direction pinched by two positive constants. Such restriction allows us to obtain sharper estimates for the width R− r than in [3]. Furthermore, for such surfaces we are able to derive an upper bound on the quotient R/r, which can be arbitrarily large for a hypersurface with normal curvatures just bounded below. Let us denote by M(c) a complete simply connected m-dimensional Riemannian manifold of constant sectional curvatures equal to c. In order to state the main results, we need the following de nition. De nition 1.1. A hypersurface F ⊂ M(c) is said to be κ1, κ2-convex (with κ2 > κ1, and for c = 0 we assume that κ1 > 0, for c > 0 we assume that κ1 > 0 and for c √ −c), if for any point P ∈ F there exist two nested geodesic spheres S2 ⊂ S1 ⊂ M(c) of constant normal curvatures equal to, respectively, κ1 and κ2, and passing through P such that locally near P the hypersurface F lies inside S1 and outside S2. 2010 Mathematics Subject Classi cation. 53C40.

As a contribution to a tentative formulation of atomic physics in a curved space, the determination of atomic fine structure energies in a space of constant curvature is investigated. Starting from the Dirac equation in a curved space-time, the analogue of the Pauli equation in a general coordinate system is derived. When particularising these results to the model of a spherical three-space with a Coulombic field, one obtains the 'curved' form of the one-electron fine structure Hamiltonian, i.e. the curved form of the Lande spin-orbit interaction and of the relativistic correction of the kinetic energy as well as some additional terms which vanish at the traditional flat limit. The theoretical curvature induced shifts and splittings of the fine structure energy levels are put in evidence and examined for the particular case of the hydrogenic n=2 levels.

Hypersurfaces with parallel Ricci tensor in spaces of constant curvature are classified. The main tool is a generalization of Moore’s decomposition theorem for isometric immersions.

1. In [3] the author has obtained some formulas for the Laplacians of higher fundamental tensors of a minimal surface in a space of constant curvature. The purpose of this paper is to give a generalization of [3] in the case of higher dimensional submanifolds. In §2, we define higher order fundamental tensors and introduce a concept of higher order normal connections. Moreover higher order Codazzi equations are proved under the assumption that the ambiant manifold is of constant curvature. In § 3 we shall calculate the 2nd commutation formula of a higher fundamental tensor and give a formula for the Laplacians of such tensors. As an application of these results, we shall prove some theorem. I think the theory in this paper is different from the classical works of W. Mayer [4], etc. After the work has finished, the author obtained a preprint P. Dombrowski [2]. The §5 in [2] is closely related to our work. Especially the Proposition 1 in this paper may be included in the Proposition 5.6 of [2]. But I think that Propositions 2 and 3 in this paper are new.