Mathematics – Number Theory
Scientific paper
2003-02-10
C. R. Acad. Sci. Paris, Ser. I 336 (2003) 919-924
Mathematics
Number Theory
8 pages, with a summary in English. One or two things added
Scientific paper
10.1016/S1631-073X(03)00223-1
Dyson a associe aux determinants de Fredholm des noyaux de Dirichlet pairs (resp. impairs) une equation de Schrodinger sur un demi-axe et a employe les methodes du scattering inverse de Gel'fand-Levitan et de Marchenko, en tandem, pour etudier l'asymptotique de ces determinants. Nous avons propose suite a notre mise-au-jour de l'operateur conducteur de chercher a realiser la transformation de Fourier elle-meme comme un scattering, et nous obtenons ici dans ce but deux systemes de Dirac sur l'axe reel tout entier et qui sont associes intrinsequement, respectivement, aux transformations en cosinus et en sinus. (Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrodinger equation on the half-axis and has used, in tandem, the Gel'fand-Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms.)
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