Non-perturbative calculations for the effective potential of the PTPT symmetric and non-Hermitian (−gϕ4)(-g\phi^{4}) field theoretic model

We investigate the effective potential of the $PT$ symmetric $(-g\phi^{4})$ field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential from which the predicted vacuum condensate vanishes exponentially as $G\to G^+$ in agreement with previous calculations. For the higher orders, we employed the invariance of the bare parameters under the change of the mass scale $t$ to fix the transformed form totally equivalent to the original theory. The form so obtained up to $G^3$ is new and shows that all the 1PI amplitudes are perurbative for both $G\ll 1$ and $G\gg 1$ regions. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all $G$ values. This unique formula is necessary because the effective potential is the generating functional for all the 1PI amplitudes which can be obtained via $\partial^n E/\partial b^n$ and thus we can obtain an analytic calculation for the 1PI amplitudes. Again, the resummed from of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit of previous calculation concerning bound states.Comment: 20 page

Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relics

This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory, which makes use of asymptotic matching, and the delta expansion, which is a powerful technique for solving nonlinear differential equations. Two different Boltzmann equations are considered. The first is derived from general relativistic considerations and the second arises in dilatonic string cosmology. The global asymptotic analysis presented here is used to find the long-time behavior of the solutions to these equations. In the first case the nature of the so-called freeze-out region and the post-freeze-out behavior is explored. In the second case the effect of the dilaton on cold dark-matter abundances is calculated and it is shown that there is a large-time power-law fall off of the dark-matter abundance. Corrections to the power-law behavior are also calculated.Comment: 15 pages, no 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

A Discontinuity in the Distribution of Fixed Point Sums

The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this discontinuity and study the possible existence of other discontinuities in $f(n,r)$ for permutations. We generalize our results to other families of structures that exhibit the same kind of discontinuities, by studying $f(n,r)$ when ``fixed points'' is replaced by ``components of size 1'' in a suitable graph of the structure. Among the objects considered are permutations, all functions and set partitions.Comment: 1 figur

A Demonstration of LISA Laser Communication

Over the past few years questions have been raised concerning the use of laser communications links between sciencecraft to transmit phase information crucial to the reduction of laser frequency noise in the LISA science measurement. The concern is that applying medium frequency phase modulations to the laser carrier could compromise the phase stability of the LISA fringe signal. We have modified the table-top interferometer presented in a previous article by applying phase modulations to the laser beams in order to evaluate the effects of such modulations on the LISA science fringe signal. We have demonstrated that the phase resolution of the science signal is not degraded by the presence of medium frequency phase modulations.Comment: minor corrections found in the CQG versio

PT-symmetry and its spontaneous breakdown explained by anti-linearity

The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner's normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, PE-symmetric Harniltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one- or two-dimensional irreducible representations of the symmetry PE. In this framework, the concept of a spontaneously broken PE-symmetry is not needed

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Homogeneity of Stellar Populations in Early-Type Galaxies with Different X-ray Properties

We have found the stellar populations of early-type galaxies are homogeneous with no significant difference in color or Mg2 index, despite the dichotomy between X-ray extended early-type galaxies and X-ray compact ones. Since the X-ray properties reflect the potential gravitational structure and hence the process of galaxy formation, the homogeneity of the stellar populations implies that the formation of stars in early-type galaxies predat es the epoch when the dichotomy of the potential structure was established.Comment: 6 pages, 5 figures, accepted for publication in Ap

Direct Measurement of intermediate-range Casimir-Polder potentials

We present the first direct measurements of Casimir-Polder forces between solid surfaces and atomic gases in the transition regime between the electrostatic short-distance and the retarded long-distance limit. The experimental method is based on ultracold ground-state Rb atoms that are reflected from evanescent wave barriers at the surface of a dielectric glass prism. Our novel approach does not require assumptions about the potential shape. The experimental data confirm the theoretical prediction in the transition regime.Comment: 4 pages, 3 figure

Polynomial solutions of nonlinear integral equations

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.Comment: 10 page