Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

Correlated exponential functions in high precision calculations for diatomic molecules

Various properties of the general two-center two-electron integral over the explicitly correlated exponential function are analyzed for the potential use in high precision calculations for diatomic molecules. A compact one dimensional integral representation is found, which is suited for the numerical evaluation. Together with recurrence relations, it makes possible the calculation of the two-center two-electron integral with arbitrary powers of electron distances. Alternative approach via the Taylor series in the internuclear distance is also investigated. Although numerically slower, it can be used in cases when recurrences lose stability. Separate analysis is devoted to molecular integrals with integer powers of interelectronic distances $r_{12}$ and the vanishing corresponding nonlinear parameter. Several methods of their evaluation are proposed.Comment: 26 pages, includes two tables with exemplary calculation

On the distribution of the nodal sets of random spherical harmonics

We study the length of the nodal set of eigenfunctions of the Laplacian on the \spheredim-dimensional sphere. It is well known that the eigenspaces corresponding to \eigval=n(n+\spheredim-1) are the spaces \eigspc of spherical harmonics of degree $n$, of dimension \eigspcdim. We use the multiplicity of the eigenvalues to endow \eigspc with the Gaussian probability measure and study the distribution of the \spheredim-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to \sqrt{\eigval}. One of our main results is bounding the variance of the volume to be O(\frac{\eigval}{\sqrt{\eigspcdim}}). In addition to the volume of the nodal set, we study its Leray measure. For every $n$, the expected value of the Leray measure is $\frac{1}{\sqrt{2\pi}}$. We are able to determine that the asymptotic form of the variance is \frac{const}{\eigspcdim}.Comment: 47 pages, accepted for publication in the Journal of Mathematical Physics. Lemmas 2.5, 2.11 were proved for any dimension, some other, suggested by the referee, modifications and corrections, were mad

Statistical physics of power fluctuations in mode locked lasers

We present an analysis of the power fluctuations in the statistical steady state of a passively mode locked laser. We use statistical light-mode theory to map this problem to that of fluctuations in a reference equilibrium statistical physics problem, and in this way study the fluctuations non-perturbatively. The power fluctuations, being non-critical, are Gaussian and proportional in amplitude to the inverse square root of the number of degrees of freedom. We calculate explicit analytic expressions for the covariance matrix of the overall, pulse and cw power variables, providing complete information on the single-time power distribution in the laser, and derive a set of fluctuation-dissipation relations between them and the susceptibilities of the steady-state quantities.Comment: 7 pages, 1 figure, RevTe

Nearsightedness of Electronic Matter in One Dimension

The concept of nearsightedeness of electronic matter (NEM) was introduced by W. Kohn in 1996 as the physical principal underlining Yang's electronic structure alghoritm of divide and conquer. It describes the fact that, for fixed chemical potential, local electronic properties at a point $r$, like the density $n(r)$, depend significantly on the external potential $v$ only at nearby points. Changes $\Delta v$ of that potential, {\it no matter how large}, beyond a distance $\textsf{R}$, have {\it limited} effects on local electronic properties, which tend to zero as function of $\textsf{R}$. This remains true even if the changes in the external potential completely surrounds the point $r$. NEM can be quantitatively characterized by the nearsightedness range, $\textsf{\textsf{R}}(r,\Delta n)$, defined as the smallest distance from $r$, beyond which {\it any} change of the external potential produces a density change, at $r$, smaller than a given $\Delta n$. The present paper gives a detailed analysis of NEM for periodic metals and insulators in 1D and includes sharp, explicit estimates of the nearsightedness range. Since NEM involves arbitrary changes of the external potential, strong, even qualitative changes can occur in the system, such as the discretization of energy bands or the complete filling of the insulating gap of an insulator with continuum spectrum. In spite of such drastic changes, we show that $\Delta v$ has only a limited effect on the density, which can be quantified in terms of simple parameters of the unperturbed system.Comment: 10 pages, 9 figure

Full-analytic frequency-domain 1pN-accurate gravitational wave forms from eccentric compact binaries

The article provides ready-to-use 1pN-accurate frequency-domain gravitational wave forms for eccentric nonspinning compact binaries of arbitrary mass ratio including the first post-Newtonian (1pN) point particle corrections to the far-zone gravitational wave amplitude, given in terms of tensor spherical harmonics. The averaged equations for the decay of the eccentricity and growth of radial frequency due to radiation reaction are used to provide stationary phase approximations to the frequency-domain wave forms.Comment: 28 pages, submitted to PR

High-density correlation energy expansion of the one-dimensional uniform electron gas

We show that the expression of the high-density (i.e small-$r_s$) correlation energy per electron for the one-dimensional uniform electron gas can be obtained by conventional perturbation theory and is of the form \Ec(r_s) = -\pi^2/360 + 0.00845 r_s + ..., where $r_s$ is the average radius of an electron. Combining these new results with the low-density correlation energy expansion, we propose a local-density approximation correlation functional, which deviates by a maximum of 0.1 millihartree compared to the benchmark DMC calculations.Comment: 7 pages, 2 figures, 3 tables, accepted for publication in J. Chem. Phy

Discrete diffraction and shape-invariant beams in optical waveguide arrays

General properties of linear propagation of discretized light in homogeneous and curved waveguide arrays are comprehensively investigated and compared to those of paraxial diffraction in continuous media. In particular, general laws describing beam spreading, beam decay and discrete far-field patterns in homogeneous arrays are derived using the method of moments and the steepest descend method. In curved arrays, the method of moments is extended to describe evolution of global beam parameters. A family of beams which propagate in curved arrays maintaining their functional shape -referred to as discrete Bessel beams- is also introduced. Propagation of discrete Bessel beams in waveguide arrays is simply described by the evolution of a complex $q$ parameter similar to the complex $q$ parameter used for Gaussian beams in continuous lensguide media. A few applications of the $q$ parameter formalism are discussed, including beam collimation and polygonal optical Bloch oscillations. \Comment: 14 pages, 5 figure

Vector and tensor perturbations in Horava-Lifshitz cosmology

We study cosmological vector and tensor perturbations in Horava-Lifshitz gravity, adopting the most general Sotiriou-Visser-Weinfurtner generalization without the detailed balance but with projectability condition. After deriving the general formulas in a flat FRW background, we find that the vector perturbations are identical to those given in general relativity. This is true also in the non-flat cases. For the tensor perturbations, high order derivatives of the curvatures produce effectively an anisotropic stress, which could have significant efforts on the high-frequency modes of gravitational waves, while for the low-frenquency modes, the efforts are negligible. The power spectrum is scale-invariant in the UV regime, because of the particular dispersion relations. But, due to lower-order corrections, it will eventually reduce to that given in GR in the IR limit. Applying the general formulas to the de Sitter and power-law backgrounds, we calculate the power spectrum and index, using the uniform approximations, and obtain their analytical expressions in both cases.Comment: Correct some typos and add new references. Version to be published in Physical Reviews

Loop Corrections in the Spectrum of 2D Hawking Radiation

We determine the one-loop and the two-loop back-reaction corrections in the spectrum of the Hawking radiation for the CGHS model of 2d dilaton gravity by evaluating the Bogoliubov coefficients for a massless scalar field propagating on the corresponding backgrounds. Since the back-reaction can induce a small shift in the position of the classical horizon, we find that a positive shift leads to a non-Planckian late-time spectrum, while a null or a negative shift leads to a Planckian late-time spectrum in the leading-order stationary-point approximation. In the one-loop case there are no corrections to the classical Hawking temperature, while in the two-loop case the temperature is three times greater than the classical value. We argue that these results are consistent with the behaviour of the Hawking flux obtained from the operator quantization only for the times which are not too late, in accordance with the limits of validity of the semiclassical approximation.Comment: 20 pages, latex, no figure