Continuum Singularities of a Mean Field Theory of Collisions

Consider a complex energy $z$ for a $N$-particle Hamiltonian $H$ and let $\chi$ be any wave packet accounting for any channel flux. The time independent mean field (TIMF) approximation of the inhomogeneous, linear equation $(z-H)|\Psi>=|\chi>$ consists in replacing $\Psi$ by a product or Slater determinant $\phi$ of single particle states $\phi_i.$ This results, under the Schwinger variational principle, into self consistent TIMF equations $(\eta_i-h_i)|\phi_i>=|\chi_i>$ in single particle space. The method is a generalization of the Hartree-Fock (HF) replacement of the $N$-body homogeneous linear equation $(E-H)|\Psi>=0$ by single particle HF diagonalizations $(e_i-h_i)|\phi_i>=0.$ We show how, despite strong nonlinearities in this mean field method, threshold singularities of the {\it inhomogeneous} TIMF equations are linked to solutions of the {\it homogeneous} HF equations.Comment: 21 pages, 14 figure

From "Dirac combs" to Fourier-positivity

Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the "Dirac comb" distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and (radial) 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text

On the positivity of Fourier transforms

Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.Comment: 12 pages, 9 figures (in 4 groups

Elementary Derivative Tasks and Neural Net Multiscale Analysis of Tasks

Neural nets are known to be universal approximators. In particular, formal neurons implementing wavelets have been shown to build nets able to approximate any multidimensional task. Such very specialized formal neurons may be, however, difficult to obtain biologically and/or industrially. In this paper we relax the constraint of a strict ``Fourier analysis'' of tasks. Rather, we use a finite number of more realistic formal neurons implementing elementary tasks such as ``window'' or ``Mexican hat'' responses, with adjustable widths. This is shown to provide a reasonably efficient, practical and robust, multifrequency analysis. A training algorithm, optimizing the task with respect to the widths of the responses, reveals two distinct training modes. The first mode induces some of the formal neurons to become identical, hence promotes ``derivative tasks''. The other mode keeps the formal neurons distinct.Comment: latex neurondlt.tex, 7 files, 6 figures, 9 pages [SPhT-T01/064], submitted to Phys. Rev.

On positive functions with positive Fourier transforms

Using the basis of Hermite-Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive the practical constraints for a function and its Fourier transform to be both positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier-Bessel transforms and the algebra of Laguerre polynomials) and to adding constraints on derivatives, such as monotonicity or convexity.Comment: 12 pages, 23 figures. High definition figures can be obtained upon request at [email protected] or [email protected]

Existence of a Density Functional for an Intrinsic State

A generalization of the Hohenberg-Kohn theorem proves the existence of a density functional for an intrinsic state, symmetry violating, out of which a physical state with good quantum numbers can be projected.Comment: 6 page