Radial Coulomb and Oscillator Systems in Arbitrary Dimensions

A mapping is obtained relating analytical radial Coulomb systems in any dimension greater than one to analytical radial oscillators in any dimension. This mapping, involving supersymmetry-based quantum-defect theory, is possible for dimensions unavailable to conventional mappings. Among the special cases is an injection from bound states of the three-dimensional radial Coulomb system into a three-dimensional radial isotropic oscillator where one of the two systems has an analytical quantum defect. The issue of mapping the continuum states is briefly considered.Comment: accepted for publication in J. Math. Phy

Singular solutions to the Seiberg-Witten and Freund equations on flat space from an iterative method

Although it is well known that the Seiberg-Witten equations do not admit nontrivial $L^2$ solutions in flat space, singular solutions to them have been previously exhibited -- either in $R^3$ or in the dimensionally reduced spaces $R^2$ and $R^1$ -- which have physical interest. In this work, we employ an extension of the Hopf fibration to obtain an iterative procedure to generate particular singular solutions to the Seiberg-Witten and Freund equations on flat space. Examples of solutions obtained by such method are presented and briefly discussed.Comment: 7 pages, minor changes. To appear in J. Math. Phy

Unified treatment of the Coulomb and harmonic oscillator potentials in DD dimensions

Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The $D$ dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calculate the Green's operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the one-dimensional case too, and point out that the complications arising due to the singularity of the one-dimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.Comment: 18 pages, 3 ps figures, revte

An Invertible Linearization Map for the Quartic Oscillator

The set of world lines for the non-relativistic quartic oscillator satisfying Newton's equation of motion for all space and time in 1-1 dimensions with no constraints other than the "spring" restoring force is shown to be equivalent (1-1-onto) to the corresponding set for the harmonic oscillator. This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature. In the context stated, the map also explicitly solves Newton's equation for the quartic oscillator for arbitrary initial data on the real line. This map is extended to all attractive potentials given by even powers of the space coordinate. It thus provides classes of new solutions to the initial value problem for all these potentials

Multiple electron trapping in the fragmentation of strongly driven molecules

We present a theoretical quasiclassical study of the formation, during Coulomb explosion, of two highly excited neutral H atoms (double H$^{*}$) of strongly driven H$_2$. In this process, after the laser field is turned off each electron occupies a Rydberg state of an H atom. We show that two-electron effects are important in order to correctly account for double H$^{*}$ formation. We find that the route to forming two H$^{*}$ atoms is similar to pathway B that was identified in Phys. Rev. A {\bf 85} 011402 (R) as one of the two routes leading to single H$^{*}$ formation. However, instead of one ionization step being "frustrated" as is the case for pathway B, both ionization steps are "frustrated" in double H$^{*}$ formation. Moreover, we compute the screened nuclear charge that drives the explosion of the nuclei during double H$^{*}$ formation.Comment: 4 pages, 6 figure

Coulomb-oscillator duality in spaces of constant curvature

In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum mechanical analogues of the Kepler-Coulomb problem on these spheres are discussed.Comment: 33 pages, LaTeX fil

N-Body Growth of a Bahcall-Wolf Cusp Around a Black Hole

We present a clear N-body realization of the growth of a Bahcall-Wolf f ~ E^0.25 (rho ~ 1/r^1.75) density cusp around a massive object ("black hole") at the center of a stellar system. Our N-body algorithm incorporates a novel implementation of Mikkola-Aarseth chain regularization to handle close interactions between star and black hole particles. Forces outside the chain were integrated on a GRAPE-6A/8 special-purpose computer with particle numbers up to N=0.25 x 10^6. We compare our N-body results with predictions of the isotropic Fokker-Planck equation and verify that the time dependence of the density (both configuration and phase-space) predicted by the Fokker-Planck equation is well reproduced by the N-body algorithm, for various choices of N and of the black hole mass. Our results demonstrate the feasibility of direct-force integration techniques for simulating the evolution of galactic nuclei on relaxation time scales.Comment: 4 pages, 3 figure

How to relate the oscillator and Coulomb systems on spheres and pseudospheres?

We show that the oscillators on a sphere and pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere: the even states of an oscillator yields the conventional Coulomb system on pseudosphere, while the odd states yield the Coulomb system on pseudosphere in the presence of magnetic flux tube generating half spin. In the higher dimensions the oscillator and Coulomb(-like) systems are connected in the similar way. In particular, applying the Kustaanheimo-Stiefel transformation to the oscillators on sphere and pseudosphere, we obtained the preudospherical generalization of MIC-Kepler problem describing three-dimensional charge-dyon system.Comment: 12 pages, Based on talk given at XXIII Colloquium on Group Theoretical Methods in Physics (July 31-August 5, 2000, Dubna

Autoparallels From a New Action Principle

We present a simpler and more powerful version of the recently-discovered action principle for the motion of a spinless point particle in spacetimes with curvature and torsion. The surprising feature of the new principle is that an action involving only the metric can produce an equation of motion with a torsion force, thus changing geodesics to autoparallels. This additional torsion force arises from a noncommutativity of variations with parameter derivatives of the paths due to the closure failure of parallelograms in the presence of torsionComment: Paper in src. Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html Read paper directly with Netscape under http://www.physik.fu-berlin.de/~kleinert/kleiner_re243/preprint.htm

Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits

A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori and thereby allows one to characterize the periodic orbits by a set of winding numbers. With this knowledge, we construct the action variables as functions of the frequency ratios and carry out a semiclassical torus quantization. The semiclassical energy levels thus obtained agree well with exact quantum calculations