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

Attosecond time-scale multi-electron collisions in the Coulomb four-body problem: traces in classical probability densities

In the triple ionization of the Li ground state by single photon absorption the three electrons escape to the continuum mainly through two collision sequences with individual collisions separated by time intervals on the attosecond scale. We investigate the traces of these two collision sequences in the classical probability densities. We show that each collision sequence has characteristic phase space properties which distinguish it from the other. Classical probability densities are the closest analog to quantum mechanical densities allowing our results to be directly compared to quantum mechanical results.Comment: 9 pages, 10 figure

Quantum Oscillator on \DC P^n in a constant magnetic field

We construct the quantum oscillator interacting with a constant magnetic field on complex projective spaces \DC P^N, as well as on their non-compact counterparts, i. e. the $N-$dimensional Lobachewski spaces ${\cal L}_N$. We find the spectrum of this system and the complete basis of wavefunctions. Surprisingly, the inclusion of a magnetic field does not yield any qualitative change in the energy spectrum. For $N>1$ the magnetic field does not break the superintegrability of the system, whereas for N=1 it preserves the exact solvability of the system. We extend this results to the cones constructed over \DC P^N and ${\cal L}_N$, and perform the (Kustaanheimo-Stiefel) transformation of these systems to the three-dimensional Coulomb-like systems.Comment: 9 pages, 1 figur

The Lorentz group and its finite field analogues: local isomorphism and approximation

Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real 4-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary accuracy to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields

Semiclassical initial value calculations of collinear helium atom

Semiclassical calculations using the Herman-Kluk initial value treatment are performed to determine energy eigenvalues of bound and resonance states of the collinear helium atom. Both the $eZe$ configuration (where the classical motion is fully chaotic) and the $Zee$ configuration (where the classical dynamics is nearly integrable) are treated. The classical motion is regularized to remove singularities that occur when the electrons collide with the nucleus. Very good agreement is obtained with quantum energies for bound and resonance states calculated by the complex rotation method.Comment: 24 pages, 3 figures. Submitted to J. Phys.

Recoil collisions as a portal to field assisted ionization at near-UV frequencies in the Strong Field Double Ionization of Helium

We explore the dependence of the double ionization of the He atom on the frequency of a strong laser field while keeping the ponderomotive energy constant. As we increase the frequency we find that the remarkable "finger-like" structure for high momenta recently found for $\omega=0.055$ a.u. \cite{Staudte, Rudenko} persists for higher frequencies. At the same time, at $\omega=0.187$ a.u. a new X-shape structure emerges for small momenta that prevails in the correlated momenta distribution. The role of this structure as a signature of the frequency dependence of non-sequential double ionization is discussed

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