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A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton-Jacobi and Schroedinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined., 27 pages

Given any space-time $M$ without singularities and any event $O$, there is a natural continuous mapping $f$ of a two dimensional sphere into any space-like slice $T$ not containing $O$. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere $S^2$ and the map $f$ sends a point in $S^2$ to the point in $T$ which is the intersection of the corresponding geodesic with $T$. To require that $f$, which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of $f$ has an odd number of preimages, is to place a very strong condition on $f$. This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on $f$ is probably too restrictive to occur in general; and if it appears to hold in a specific example, then some $f$ should be calculated either analytically or numerically to provide either an illustrative example or counterexample., 8 pages, amstex

Representations of the quantum algebra suq(2) are constructed in terms of q‐special functions, defined on real O(3) spheres and real planes. They include q‐Vilenkin functions, related to little q‐Jacobi functions, q‐spherical functions, and q‐Legendre polynomials.

Dynamical invariants for time‐dependent (TD) oscillator systems in two dimensions are derived. The applications of the dynamical algebraic approach known already for one‐dimensional TD systems [R. S. Kaushal and H. J. Korsch, J. Math. Phys. 22, 1904 (1981)] are now carried out to two‐dimensional TD systems. A possible generalization of Ermakov systems to higher dimensions is discussed.

The existence of Killing vectors and Killing tensors in two‐dimensional space‐times is discussed. The corresponding mechanical problem having a potential with two exponential terms is analyzed in detail and a number of cases admitting Killing tensors are given. The Killing tensors give rise to new exact solutions in perfect fluid Bianchi and Kantowski–Sachs cosmologies as well as in inflationary models with a scalar field source.

A space-time may be regarded as a partially ordered set by causal relations. Recently, as a poset, a sphere order representation of a space-time is considered and some interesting results are obtained. In this paper, we define a weakly causally convex open set and characterize connected open subsets of the n-dimensional Minkowski space-time which admit a continuous n-sphere order representation. Then, some necessary and sufficient conditions for admitting a continuous two-sphere order representation of two-dimensional space-times, are given. As a result, it is shown that any simply connected and causally simple space-time of dimension two can be causally isomorphically imbedded into the two-dimensional Minkowski space-time.

In this paper, we prove the global existence of classical solutions to the Cauchy problem for the minimal surface equation in the Minkowski space R1+2 with slow decay initial value.

In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low...

The path space measures that have been constructed in the present authors’ previous papers to give path integral formulas in quantum mechanics for a Dirac particle in two‐dimensional space‐time are shown to be concentrated on those paths that have differential coefficients equal to plus or minus the light velocity in every finite time interval except at finitely many instants of time.

Necessary and sufficient conditions are derived for the analytic extendability of a static two‐dimensional space‐time. For all allowed cases, explicit analytic extensions are determined together with their corresponding Penrose‐Carter diagrams. Extensions are classified and further discussed in terms of these diagrams, with special consideration given to the question of bifurcate Killing horizons. The application of these results to four‐dimensional relativistic space‐times is illustrated with specific examples.