The two-dimensional hydrogen atom revisited

The bound state energy eigenvalues for the two-dimensional Kepler problem are found to be degenerate. This "accidental" degeneracy is due to the existence of a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schroedinger equation. This equation is solved by projecting the two-dimensional momentum space onto the surface of a three-dimensional sphere. The eigenfunctions are then expanded in terms of spherical harmonics, and this leads to an integral relation in terms of special functions which has not previously been tabulated. The dynamical symmetry of the problem is also considered, and it is shown that the two components of the Runge-Lenz vector in real space correspond to the generators of infinitesimal rotations about the respective coordinate axes in momentum space.Comment: 10 pages, no figures, RevTex

Parity-dependent squeezing of light

A parity-dependent squeezing operator is introduced which imposes different SU(1,1) rotations on the even and odd subspaces of the harmonic oscillator Hilbert space. This operator is used to define parity-dependent squeezed states which exhibit highly nonclassical properties such as strong antibunching, quadrature squeezing, strong oscillations in the photon-number distribution, etc. In contrast to the usual squeezed states whose $Q$ and Wigner functions are simply Gaussians, the parity-dependent squeezed states have much more complicated $Q$ and Wigner functions that exhibit an interesting interference in phase space. The generation of these states by parity-dependent quadratic Hamiltonians is also discussed.Comment: accepted for publication in J. Phys. A, LaTeX, 11 pages, 12 figures (compressed PostScript, available at http://www.technion.ac.il/~brif/graphics/pdss_graph ). More information on http://www.technion.ac.il/~brif/science.htm

Dressing a black hole with non-minimally coupled scalar field hair

We investigate the possibility of dressing a four-dimensional black hole with classical scalar field hair which is non-minimally coupled to the space-time curvature. Our model includes a cosmological constant but no self-interaction potential for the scalar field. We are able to rule out black hole hair except when the cosmological constant is negative and the constant governing the coupling to the Ricci scalar curvature is positive. In this case, non-trivial hairy black hole solutions exist, at least some of which are linearly stable. However, when the coupling constant becomes too large, the black hole hair becomes unstable.Comment: 17 pages, 7 figures, uses iopart.cls. Minor changes, accepted for publication in Classical and Quantum Gravit

(2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase

A particle system with a (2+1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and supercritical phases (describing 2D isotropic ordinary and exotic oscillators) are separated by the critical phase (one-mode oscillator), and are related by a duality transformation. In the flat limit, the system transforms into a free Galilean exotic particle on the noncommutative plane. The wave equations carrying projective representations of the exotic Newton-Hooke symmetry are constructed.Comment: 30 pages, 2 figures; typos correcte

Equivalence of the Siegert-pseudostate and Lagrange-mesh R-matrix methods

Siegert pseudostates are purely outgoing states at some fixed point expanded over a finite basis. With discretized variables, they provide an accurate description of scattering in the s wave for short-range potentials with few basis states. The R-matrix method combined with a Lagrange basis, i.e. functions which vanish at all points of a mesh but one, leads to simple mesh-like equations which also allow an accurate description of scattering. These methods are shown to be exactly equivalent for any basis size, with or without discretization. The comparison of their assumptions shows how to accurately derive poles of the scattering matrix in the R-matrix formalism and suggests how to extend the Siegert-pseudostate method to higher partial waves. The different concepts are illustrated with the Bargmann potential and with the centrifugal potential. A simplification of the R-matrix treatment can usefully be extended to the Siegert-pseudostate method.Comment: 19 pages, 1 figur

Magnetic von-Neumann lattice for two-dimensional electrons in the magnetic field

One-particle eigenstates and eigenvalues of two-dimensional electrons in the strong magnetic field with short range impurity and impurities, cosine potential, boundary potential, and periodic array of short range potentials are obtained by magnetic von-Neumann lattice in which Landau level wave functions have minimum spatial extensions. We find that there is a dual correspondence between cosine potential and lattice kinetic term and that the representation based on the von-Neumann lattice is quite useful for solving the system's dynamics.Comment: 21pages, figures not included, EPHOU-94-00

Bogoliubov transformations and exact isolated solutions for simple non-adiabatic Hamiltonians

We present a new method for finding isolated exact solutions of a class of non-adiabatic Hamiltonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.Comment: LaTeX, 16 pages, 1 figure. Minor additions and journal re

Jones-matrix Formalism as a Representation of the Lorentz Group

It is shown that the two-by-two Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. The attenuation and phase-shift filters are represented respectively by the three-parameter rotation subgroup and the three-parameter Lorentz group for two spatial and one time dimensions. It is noted that the Lorentz group has another three-parameter subgroup which is like the two-dimensional Euclidean group. Possible optical filters having this Euclidean symmetry are discussed in detail. It is shown also that the Jones-matrix formalism can be extended to some of the non-orthogonal polarization coordinate systems within the framework of the Lorentz-group representation.Comment: RevTeX, 27 pages, no figures, to be published in J. Opt. Soc. Am.