Green’s functions and strength functions for stationary quantum systems

By using the analytic properties of the retarded Green’s function for a stationary quantum system, the strength function that coincides with the energy distribution of an unperturbed state of the system over its exact states in a perturbing field is constructed. It is shown that, in general, this strength function has the form of a Breit-Wigner distribution with energy-dependent parameters and that its moments are determined by the expectation values of various powers of the exact Hamiltonian for the wave function of the unperturbed state. The strength function averaged over a certain energy interval is calculated, and its properties are investigated for a global regime of averaging. The resulting strength functions are used to determine the mean field and the optical potential for nucleons in nuclei and to investigate conditions under which quantum chaos emerges in various systems.     

Three-dimensional covariant π-meson wave functions

A system of relativistic three-dimensional equations of quasipotential type is obtained for -meson wave functions, the pseudoscalar and pseudovector spinorial structures taken into account.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 100–104, April 1984.The author thanks V. G. Kadyshevskii and N. B. Skachtov for their interest to this work and for their useful discussions of the results.     

Saturation functions of non-linear standing waves

1IntroductionItwasfoundthatalltheharmonicsofanon-linearstandingwaveexhibitsaturationten-dency'esPeciallysignificalltasthefundamentalsoundpressureisraisedoverl5odBre2OpPaor'so,andtheexperimentallawofsaturationwasobtainedbycurve-fittingll].Theformulasobtainedagreedwellwiththeexperimentaldata,within1dBinthewholerange.Theexper-imeotswerecarriedonfundamelltalfrequencies22OHz,294Hzand366Hz,thethird,fourthandforhresonancefrequenciesoftheclosedtube,respectively,denotedbym=3,tn=4,andm=5.Theorderoft…     

Connection formulae for painlevé V functions

We study those Painlevé V equations which admit a one-parameter family of solutions analytic at the origin and solve the global problem of connecting the two-parameter asymptotic expansion at t → ∞ with the two-parameter expansion about t = 0. This connection is the one needed to study the correlation functions of the transverse Ising chain at the critical field strength. From these connection formulae, we derive a one-parameter connection formula for the general Painlevé II equation and a two-parameter connection formula for a class of Painlevé III equations.     

Must thermodynamic functions be piecewise analytic?

There seems to be no general theoretical argument to support the idea that thermodynamic functions are piecewise analytic. We suggest that nonanalyticity may be associated with Gibbs states which are quasiperiodic under space translations, or have a more general nonperiodic (turbulent) behavior.     

Phase behaviors of transfer functions in Vibrating systems

1IlltroductionTransferfunction(TF)possessesimportantinformationregardingthetransmissionpathsinvibrationandnoisecontrol.Thisinformationcanbeusedinthediagnosticsofvibratingsystem,therecoveryofsourcewateform,andthecontrolofsoundfieldsl1'2].TheaverageresponseofasystemisdeterminedmainlybythemagnitudeoftheTF.However,theincreaseduseofacousticalandvibrationalsignalsfordiagnosticsofmachineryhasletresearchersreconsiderthephasepropertiesofTFs.Thephase,inparticular,isveryimportantifonewantstorecoverthe…     

New functions of VCSEL-based optical devices Invited Paper

We have seen a lot of unique features off vertical cavity surface emitting lasers(VCSELs),such as low power consumption,wafer-level testing,small packaging capability,and so on.The market of VCSELs has been growing up rapidly in recent years and they are now the key devices in local area networks using multi-mode optical fibers.In addition,new functions on VCSELs have been demonstrated.In this paper, the recent advances of VCSEL photonics will be reviewed,which include the wavelength engineering and the athermal operation based on microelectro mechanical system(MEMS)technologies.Also,this paper explores the potential and challenges for new functions of VCSELs,including high-speed control of optical phase,slow light devices,plasmonic VCSELs,and so on.     

Milnor–Selberg zeta functions and zeta regularizations

By a similar idea for the construction of Milnor’s gamma functions, we introduce “higher depth determinants” of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor–Selberg zeta function. It is shown that the Milnor–Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.     

Luminosity functions forW ± andZ 0 initiated processes

We presentW/Z-parton luminosities 1/s d L/dτ in the TeV energy region for suche ⁺ e ^?,ep andpp collision processes where the intermediate vector bosonsW ^± orZ ⁰ are in the initial state.     

The design of the apodizing functions for guided-wave EOTF

A design method of the apodizing functions for guided-wave four-port electrooptic tunable filter (EOTF) is proposed. As an example, the specific form of Gaussian apodizing function is designed by analyzing the crosstalk performance of EOTF. Crosstalk is calculated based on the transient response of the filter to an input optical pulse. It is concluded that crosstalk level below -30 dB for 20-Gb/s data rates and 100-GHz channel spacing is achievable with presently available substrate dimensions.     

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ? 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.     

Renormalisation group and two-point functions in bosonic σ models

The renormalisation group equations for two-point functions involving the energy-momentum tensor in two-dimensional non-linear σ models derived previously using dimensional regularisation are solved to give the leading singular term in the operator product expansion. The results are in accord with the general framework suggested by Zamolodchikov, although the equation describing the renormalisation flow of the Virasoro central charge takes a modified form.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

Two-fold Mellin–Barnes transforms of Usyukina–Davydychev functions

In our previous paper (Allendes et al., 2013 [10]), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms. The MB transforms were written there as polynomials of logarithms of ratios of squares of the external momenta with certain coefficients. We also showed that these coefficients have a combinatoric origin. In this paper, we present an explicit formula for these coefficients. The procedure of recovering the coefficients is based on taking the double-uniform limit in certain series of smooth functions of two variables which is constructed according to a pre-determined iterative way. The result is obtained by using basic methods of mathematical analysis. We observe that the finiteness of the limit of this iterative chain of smooth functions should reflect itself in other mathematical constructions, too, since it is not related in any way to the explicit form of the MB transforms. This finite double-uniform limit is represented in terms of a differential operator with respect to an auxiliary parameter which acts on the integrand of a certain two-fold MB integral. To demonstrate that our result is compatible with original representations of UD functions, we reproduce the integrands of these original integral representations by applying this differential operator to the integrand of the simple integral representation of the scalar triangle four-dimensional integral J(1,1,1−ε)$J(1,1,1-\epsilon )$.     

On a connection between formulas about q–gamma functions

In this short communication, we want to pay attention to a few wrong formulas which are unfortunately cited and used in a dozen papers afterwards. We prove that the provided relations and asymptotic expansion about the q-gamma function are not correct. This is illustrated by numerous concrete counterexamples. The error came from the wrong assumption about the existence of a parameter which does not depend on anything. Here, we apply a similar procedure and derive a correct formula for the q-gamma function.     

Truncated Gamow functions, α-decay and the exponential law

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an -decay model to prove general validity of the exponential decay law for periods of several lifetimes.