Zeta functions of quantum graphs

In this article we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of the star. We then extend the technique to allow any matching conditions at the center for which the Laplace operator is self-adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions. In the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the Neumann star graph. In the second half of the article we apply the zeta function to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graphs. These have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature.Comment: 32 pages, typos corrected, references adde

ORFEUS II Far-UV Spectroscopy of AM Herculis

Six high-resolution (\lambda/\Delta\lambda ~ 3000) far-UV (\lambda\lambda = 910-1210 \AA) spectra of the magnetic cataclysmic variable AM Herculis were acquired in 1996 November during the flight of the ORFEUS-SPAS II mission. AM Her was in a high optical state at the time of the observations, and the spectra reveal emission lines of O VI \lambda\lambda 1032, 1038, C III \lambda 977, \lambda 1176, and He II \lambda 1085 superposed on a nearly flat continuum. Continuum flux variations can be described as per Gansicke et al. by a ~ 20 kK white dwarf with a ~ 37 kK hot spot covering a fraction f~0.15 of the surface of the white dwarf, but we caution that the expected Lyman absorption lines are not detected. The O VI emission lines have narrow and broad component structure similar to that of the optical emission lines, with radial velocities consistent with an origin in the irradiated face of the secondary and the accretion funnel, respectively. The density of the narrow- and broad-line regions is n_{nlr} ~ 3\times 10^{10} cm^{-3} and n_{blr} ~ 1\times 10^{12} cm^{-3}, respectively, yet the narrow-line region is optically thick in the O VI line and the broad-line region is optically thin; apparently, the velocity shear in the broad-line region allows the O VI photons to escape, rendering the gas effectively optically thin. Unexplained are the orbital phase variations of the emission-line fluxes.Comment: 15 pages, 6 Postscript figures; LaTeX format, uses aaspp4.sty; table2.tex included separately because it must be printed sideways - see instructions in the file; accepted on April 17, 1998 for publication in The Astrophysical Journa

The Lagrange and Markov spectra from the dynamical point of view

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).Comment: 27 pages, 6 figures. Survey articl

Properties of the series solution for Painlevé I

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented

ORFEUS II and IUE Spectroscopy of EX Hydrae

Using ORFEUS-SPAS II FUV spectra, IUE UV spectra, and archival EUVE deep survey photometry, we present a detailed picture of the behavior of the magnetic cataclysmic variable EX Hydrae. Like HUT spectra of this source, the FUV and UV spectra reveal broad emission lines of He II, C II-IV, N III and V, O VI, Si III-IV, and Al III superposed on a continuum which is blue in the UV and nearly flat in the FUV. Like ORFEUS spectra of AM Her, the O VI doublet is resolved into broad and narrow emission components. Consistent with its behavior in the optical, the FUV and UV continuum flux densities, the FUV and UV broad emission line fluxes, and the radial velocity of the O VI broad emission component all vary on the spin phase of the white dwarf, with the maximum of the FUV and UV continuum and broad emission line flux light curves coincident with maximum blueshift of the broad O VI emission component. On the binary phase, the broad dip in the EUV light curve is accompanied by strong eclipses of the UV emission lines and by variations in both the flux and radial velocity of the O VI narrow emission component. The available data are consistent with the accretion funnel being the source of the FUV and UV continuum and the O VI broad emission component, and the white dwarf being the source of the O VI narrow emission component.Comment: 21 pages, 10 Postscript figures; LaTeX format, uses aaspp4.sty; table2.tex included separately because it must be printed sideways - see instructions in the file; accepted on 1999 Feb 20 for publication in The Astrophysical Journa

Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds

We study correlation functions of single-cycle chiral operators in the symmetric product orbifold of N supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single-valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under OPEs. We then discuss extremal n-point correlators, i.e. correlators of n-1 chiral and one anti-chiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.Comment: 36 pages, 3 figures, v2: minor improvement

Induced measures in the space of mixed quantum states

We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x K mixed states, if the reduction takes place with respect to the first subsystem). For K=N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of $N \times N$ pure states behaves as lnN-1/2.Comment: 12 latex pages, 2 figures in epsf, submited to J. Phys. A. ver.3, some improvements and a few references adde

On Weierstra{\ss} semigroups at one and two points and their corresponding Poincar\'e series

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.Comment: Beginning of Section 3 and Subsection 3.1 were modifie

Dynamics of quantum entanglement

A model of discrete dynamics of entanglement of bipartite quantum state is considered. It involves a global unitary dynamics of the system and periodic actions of local bistochastic or decaying channel. For initially pure states the decay of entanglement is accompanied with an increase of von Neumann entropy of the system. We observe and discuss revivals of entanglement due to unitary interaction of both subsystems. For some mixed states having different marginal entropies of both subsystems (one of them larger than the global entropy and the other one one smaller) we find an asymmetry in speed of entanglement decay. The entanglement of these states decreases faster, if the depolarizing channel acts on the "classical" subsystem, characterized by smaller marginal entropy.Comment: 10 pages, Revtex, 10 figures, refined versio