Analytic structure of Bloch functions for linear molecular chains

This paper deals with Hamiltonians of the form H=-{\bf \nabla}^2+v(\rr), with v(\rr) periodic along the $z$ direction, $v(x,y,z+b)=v(x,y,z)$. The wavefunctions of $H$ are the well known Bloch functions \psi_{n,\lambda}(\rr), with the fundamental property $\psi_{n,\lambda}(x,y,z+b)=\lambda \psi_{n,\lambda}(x,y,z)$ and $\partial_z\psi_{n,\lambda}(x,y,z+b)=\lambda \partial_z\psi_{n,\lambda}(x,y,z)$. We give the generic analytic structure (i.e. the Riemann surface) of \psi_{n,\lambda}(\rr) and their corresponding energy, $E_n(\lambda)$, as functions of $\lambda$. We show that $E_n(\lambda)$ and $\psi_{n,\lambda}(x,y,z)$ are different branches of two multi-valued analytic functions, $E(\lambda)$ and $\psi_\lambda(x,y,z)$, with an essential singularity at $\lambda=0$ and additional branch points, which are generically of order 1 and 3, respectively. We show where these branch points come from, how they move when we change the potential and how to estimate their location. Based on these results, we give two applications: a compact expression of the Green's function and a discussion of the asymptotic behavior of the density matrix for insulating molecular chains.Comment: 13 pages, 11 figure

On the convergence of second order spectra and multiplicity

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper

Spectral analysis of temperature and Brunt-Vaisala frequency fluctuations observed by radiosondes

Recent studies have revealed that vertical wave number spectra of wind velocity and temperture fluctuations in the troposphere and the lower stratosphere are fairly well explained by a saturated gravity wave spectrum. But N(2) (N:Brunt-Vaisala (BV) frequency) spectra seem to be better for testing the scaling of the vertical wave number spectra in layers with different stratifications, beause its energy density is proportional only to the background value of N(2), while that for temperature depends on both the BV frequency and the potential temperature. From temperature profiles observed in June to August 1987 over the MU Observatory, Japan, by using a radiosonde with 30 m height resolution, N(2) spectra are determined in the 2 to 8.5 km (troposphere) and 18.5 to 25 km (lower stratosphere) ranges. Although individual spectra show fairly large day-by-day variability, the slope of the median of 34 spectra agrees reasonably with the theoretical value of -1 in the wave number range of 6 x 10(-4) similar to 3 x 10(-3) (c/m). The ratio of the spectral energy between these two height regions is about equal to the ratio of N(2), consistent with the prediction of saturated gravity wave theory

PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has a completely real spectrum for $\lambda\le 1$, and begins to develop complex eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave-functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighbouring Bloch waves.Comment: 15 pages, 7 figures Minor corrections, additional reference

Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity

We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg - de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smoothing effect up to real analyticity if the initial data only has a single point singularity at $x=0.

Adjointness Relations as a Criterion for Choosing an Inner Product

This is a contribution to the forthcoming book "Canonical Gravity: {}From Classical to Quantum" edited by J. Ehlers and H. Friedrich. Ashtekar's criterion for choosing an inner product in the quantisation of constrained systems is discussed. An erroneous claim in a previous paper is corrected and a cautionary example is presented.Comment: 6 pages, MPA-AR-94-

Asteroseismological Observations of the Central Star of the Planetary Nebula NGC 1501

We report on a global CCD time-series photometric campaign to decode the pulsations of the nucleus of the planetary nebula NGC1501. The star is hot and hydrogen-deficient, similar to the pre-white-dwarf PG 1159 stars. NGC1501 shows pulsational brightness variations of a few percent with periods ranging from 19 to 87 minutes. The variations are very complex, suggesting a pulsation spectrum that requires a long unbroken time series to resolve. Our CCD photometry of the star covers a two-week period in 1991 November, and used a global network of observatories. We obtained nearly continuous coverage over an interval of one week in the middle of the run. We have identified 10 pulsation periods, ranging from 5235 s down to 1154 s. We find strong evidence that the modes are indeed nonradial g-modes. The ratios of the frequencies of the largest-amplitude modes agree with those expected for modes that are trapped by a density discontinuity in the outer layers. We offer a model for the pulsation spectrum that includes a common period spacing of 22.3 s and a rotation period of 1.17 days; the period spacing allows us to assign a seismological mass of 0.55+/-0.03 Msun.Comment: 12 pages, AASTEX, 7 tables, 6 EPS figures, to appear in AJ, 12/96 Corrected version repairs table formatting and adds missing Table

In-the-Gap SU UMa-Type Dwarf Nova, Var73 Dra with a Supercycle of about 60 Days

An intensive photometric-observation campaign of the recently discovered SU UMa-type dwarf nova, Var73 Dra was conducted from 2002 August to 2003 February. We caught three superoutbursts in 2002 October, December and 2003 February. The recurrence cycle of the superoutburst (supercycle) is indicated to be $\sim$60 d, the shortest among the values known so far in SU UMa stars and close to those of ER UMa stars. The superhump periods measured during the first two superoutbursts were 0.104885(93) d, and 0.10623(16) d, respectively. A 0.10424(3)-d periodicity was detected in quiescence. The change rate of the superhump period during the second superoutburst was $1.7\times10^{-3}$, which is an order of magnitude larger than the largest value ever known. Outburst activity has changed from a phase of frequent normal outbursts and infrequent superoutbursts in 2001 to a phase of infrequent normal outbursts and frequent superoutbursts in 2002. Our observations are negative to an idea that this star is an related object to ER UMa stars in terms of the duty cycle of the superoutburst and the recurrence cycle of the normal outburst. However, to trace the superhump evolution throughout a superoutburst, and from quiescence more effectively, may give a fruitful result on this matter.Comment: 9 pages, 8 figures, submitted to A&

Inviscid incompressible limits of the full Navier-Stokes-Fourier system

We consider the full Navier-Stokes-Fourier system in the singular limit for the small Mach and large Reynolds and Peclet numbers, with ill prepared initial data on the three dimensional Euclidean space. The Euler-Boussinesq approximation is identified as the limit system