The quantum anharmonic oscillator in the Heisenberg picture and multiple scale techniques

Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum.Comment: 18 page

Periodic orbits for classical particles having complex energy

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.Comment: 12 pages, 9 figure

Systematics of quadrupolar correlation energies

We calculate correlation energies associated with the quadrupolar shape degrees of freedom with a view to improving the self-consistent mean-field theory of nuclear binding energies. The Generator Coordinate Method is employed using mean-field wave functions and the Skyrme SLy4 interaction. Systematic results are presented for 605 even-even nuclei of known binding energies, going from mass A=16 up to the heaviest known. The correlation energies range from 0.5 to 6.0 MeV in magnitude and are rather smooth except for large variations at magic numbers and in light nuclei. Inclusion of these correlation energies in the calculated binding energy is found to improve two deficiencies of the Skyrme mean field theory. The pure mean field theory has an exaggerated shell effect at neutron magic numbers and addition of the correlation energies reduce it. The correlations also explain the phenomenon of mutually enhanced magicity, an interaction between neutron and proton shell effects that is not explicable in mean field theory.Comment: 4 pages with 3 embedded figure

Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.Comment: Few typos have been corrected and a reference update

Entanglement Efficiencies in PT-Symmetric Quantum Mechanics

The degree of entanglement is determined for an arbitrary state of a broad class of PT-symmetric bipartite composite systems. Subsequently we quantify the rate with which entangled states are generated and show that this rate can be characterized by a small set of parameters. These relations allow one in principle to improve the ability of these systems to entangle states. It is also noticed that many relations resemble corresponding ones in conventional quantum mechanics.Comment: Published version with improved figures, 5 pages, 2 figure

Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: Complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im(E) -> 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as ReE increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure

On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.Comment: 21pages, 9 figure

Universality in Random Walk Models with Birth and Death

Models of random walks are considered in which walkers are born at one location and die at all other locations with uniform death rate. Steady-state distributions of random walkers exhibit dimensionally dependent critical behavior as a function of the birth rate. Exact analytical results for a hyperspherical lattice yield a second-order phase transition with a nontrivial critical exponent for all positive dimensions $D\neq 2,~4$. Numerical studies of hypercubic and fractal lattices indicate that these exact results are universal. Implications for the adsorption transition of polymers at curved interfaces are discussed.Comment: 11 pages, revtex, 2 postscript figure

Comparison of LISA and Atom Interferometry for Gravitational Wave Astronomy in Space

One of the atom interferometer gravitational wave missions proposed by Dimopoulos et al.1 in 2008 was called AGIS-Sat. 2. It had a suggested gravitational wave sensitivity set by the atom state detection shot noise level that started at 1 mHz, was comparable to LISA sensitivity from 1 to about 20 mHz, and had better sensitivity from 20 to 500 mHz. The separation between the spacecraft was 1,000 km, with atom interferometers 200 m long and shades from sunlight used at each end. A careful analysis of many error sources was included, but requirements on the time-stability of both the laser wavefront aberrations and the atom temperatures in the atom clouds were not investigated. After including these considerations, the laser wavefront aberration stability requirement to meet the quoted sensitivity level is about 1\times10-8 wavelengths, and is far tighter than for LISA. Also, the temperature fluctuations between atom clouds have to be less than 1 pK. An alternate atom interferometer GW mission in Earth orbit called AGIS-LEO with 30 km satellite separation has been suggested recently. The reduction of wavefront aberration noise by sending the laser beam through a high-finesse mode-scrubbing optical cavity is discussed briefly, but the requirements on such a cavity are not given. Unfortunately, such an Earth-orbiting mission seems to be considerably more difficult to design than a non-geocentric mission and does not appear to have comparably attractive scientific goals.Comment: Submitted to Proc. 46th Rencontres de Moriond: Gravitational Waves and Experimental Gravity, March 20 - 27, 2011, La Thuile, Ital

Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure x w(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.Comment: 7 pages, result generalized to include integration in the complex domai