Daniel Bessis

Pade´ approximations in noise filtering

Resume We first recall the results on the distribution of poles and zeros of Pade approximations to noisy Taylor series around a point. We then report the numerical experiment on noisy data coming from inelastic electron-scattering cross-sections on rare gases, as functions of the momentum transfer at fixed energy. We apply the Thiele algorithm (Pade approximations of type II) to those data, which are known to belong to an analytic function in a cut plane from −∞ to 0. In this case, the poles and zeros split into three families: 1. (i) Doublet pole-zero at a distance of the order of the computer roundoff. 2. (ii) Doublet pole-zero at a distance of the order of the noise in the experimental data. 3. (iii) Remaining poles and zeros compatible with the analytic structure of the experimentally measured function. By throwing away the first two families of poles and zeros (noisy ones), we were able to obtain an excellent filtering of the data compatible with the analytic structure of the analyzed function.

Universal Statistical Behavior of the Complex Zeros of Wiener Transfer Functions

To N real random variables the sample autocorrelation coefficients, which are also the N Fourier coefficients of a measure on the unit circle are associated. The polynomials orthogonal with respect to this measure define the transfer functions of the Wiener-Levinson predictors. We show that the statistics of the zeros of those random polynomials exhibits a universal law of crystallization on a circle of radius [1 - (ln N)/2n], n being the order of the predictor. These results are supported by extensive computer experiments and backed by a theoretical scaling argument in the asymptotic domain ln N??n??N. These results are independent of the nature of the noise and robust for signals of finite length N.

Design of semiconductor heterostructures with preset electron reflectance by inverse scattering techniques

Abstract We present the application of the inverse scattering method to the design of semiconductor heterostructures having a preset dependence of the (conduction) electrons’ reflectance on the energy. The electron dynamics are described by either the effective mass Schrodinger equation or by the (variable mass) BenDaniel and Duke equations. The problem of phase (re)construction for the complex transmission and reflection coefficients is solved by a combination of Pade approximation techniques, obtaining reference solutions with simple analytic properties. Reflectance-preserving transformations allow bound state and reflection resonance management. The inverse scattering problem for the Schrodinger equation is solved using an algebraic approach due to Sabatier. This solution can be mapped unitarily onto a family of BenDaniel and Duke type equations. The boundary value problem for the nonlinear equation which determines the mapping is discussed in some detail. The chemical concentration profile of heterostructures whose self consistent potential yields the desired reflectance is solved completely in the case of Schrodinger dynamics and approximately for BenDaniel and Duke dynamics. The Appendix contains a brief digest of results from the scattering and inverse scattering theory for the one-dimensional Schrodinger equation which is used in the article.

DISTRIBUTION OF ROOTS OF RANDOM REAL GENERALIZED POLYNOMIALS

AbstractThe average density of zeros for monic generalized polynomials,

Orthogonal polynomials associated to almost periodic Schro¨dinger operators: a trend towards random orthogonal polynomials

P_n (z) = \phi (z) + \sum\nolimits_{k = 1}^n {c_k f_k } (z)

Noise in the complex plane: open problems

, with real holomorphic ϕ,fk and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form

Realistic semiconductor heterostructures design using inverse scattering

P_n (z) = z^n + \sum\nolimits_{k = 1}^n {c_k z^{n - k} }