Real Zeros of the Airy Functions

1997 ◽  
pp. 421-422
Keyword(s):  
Airy Functions ◽  
Real Zeros

Derivation of tunneling probabilities for arbitrarily graded potential barriers using modified Airy functions

2010 ◽  
Vol 42 (2) ◽  
pp. 129-141
Author(s):  
Kyu-Tae Lee ◽  
Eun Joo Jung ◽  
Chul Han Kim ◽  
Chang-Min Kim
Keyword(s):  
Airy Functions ◽  
Potential Barriers ◽  
Modified Airy Functions

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

Analytical account for off-axis effects in X-ray radiation of free electron lasers

2021 ◽  
Vol 64 (1) ◽  
pp. 21-28
Author(s):  
K.V. Zhukovsky ◽  
Keyword(s):  
Free Electron ◽  
Finite Size ◽  
Photon Radiation ◽  
Beam Parameter ◽  
Beam Size ◽  
Free Electron Lasers ◽  
Airy Functions ◽  
X Ray ◽  
Harmonic Radiation ◽  
Gain Length

We give analytical description of generation of harmonics of the undulator radiation (UR) with account for the finite electron beam size, emittance, off-axis beam deviation and electron energy spread, as well as for the constant magnetic components and field harmonics effects. We give exact analytical expressions for the generalized Bessel and Airy functions, which describe the spectrum line shape and intensities in the two-frequency bi-harmonic undulator with account for the above factors. The obtained analytical formulae distinguish contributions of each field component and every undulator and beam parameter on the harmonic radiation in free electron lasers (FEL). The effect of the field on the harmonic radiation is analyzed with account for the beam finite size and its off-axis deviation. The phenomenological model is employed for the FEL modeling; with its help we study the harmonic generation, including even ones, in the experiments LCLS and LEUTL. We demonstrate analytically that strong second FEL harmonic in X-ray FEL at the wavelengths λ = 0.75nm in the LCLS experiment is caused by the deviation of the electron trajectories off the axis in 15 μm on the gain length 1.6 m, which is comparable with the beam size; the strong second FEL harmonic in the LEUTL experiment at the wavelength λ = 192nm can be attributed to interaction of the electrons in wide, ~ 0.2 mm, beam with the photon radiation at the gain length 0.87 m. The modeling results fully agree with the measurements. The developed formalism allows the analysis of projected and built FELs and their radiation, helps minimizing losses and correcting magnetic fields; it also shows physical background and reasons for each harmonic radiated power in the FEL.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals Calculating the Energy Spectrum of Complex Low-Dimensional Heterostructures in the Electric Field

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Svetlana N. Khonina ◽  
Sergey G. Volotovsky ◽  
Sergey I. Kharitonov ◽  
Nikolay L. Kazanskiy
Keyword(s):  
Steady State ◽  
Electric Field ◽  
Ground State ◽  
State Energy ◽  
Airy Functions ◽  
Piecewise Constant ◽  
Matrix Rank ◽  
The Matrix ◽  
Constant Potential ◽  
Low Dimensional

An algorithm for solving the steady-state Schrödinger equation for a complex piecewise-constant potential in the presence of theE-field is developed and implemented. The algorithm is based on the consecutive matching of solutions given by the Airy functions at the band boundaries with the matrix rank increasing by no more than two orders, which enables the characteristic solution to be obtained in the convenient form for search of the roots. The algorithm developed allows valid solutions to be obtained for the electric field magnitudes larger than the ground-state energy level, that is, when the perturbation method is not suitable.

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