The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda -soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson’s -Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum.
A l’aide de la théorie de Sato, on calcule la matrice spectrale de Stieltjes associée à une matrice de Jacobi doublement infinie, donnant lieu à une solution -soliton du réseau de Toda. On utilise ce résultat pour donner un développement explicite de la solution fondamentale de versions discrètes de l’équation de la chaleur, en termes d’une série des -déformations de Jackson des fonctions de Bessel. Pour les solitons dits de Askey-Wilson, ce développement se réduit à une somme finie.
Keywords: Heat kernel, Toda lattice hierarchy
Mot clés : noyau de la chaleur, réseau de Toda
@article{AIF_2005__55_6_1765_0, author = {Haine, Luc}, title = {The spectral matrices of {Toda} solitons and the fundamental solution of some discrete heat equations}, journal = {Annales de l'Institut Fourier}, pages = {1765--1788}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {6}, year = {2005}, doi = {10.5802/aif.2140}, mrnumber = {2187934}, zbl = {1078.35101}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2140/} }
TY - JOUR AU - Haine, Luc TI - The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations JO - Annales de l'Institut Fourier PY - 2005 SP - 1765 EP - 1788 VL - 55 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2140/ DO - 10.5802/aif.2140 LA - en ID - AIF_2005__55_6_1765_0 ER -
%0 Journal Article %A Haine, Luc %T The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations %J Annales de l'Institut Fourier %D 2005 %P 1765-1788 %V 55 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2140/ %R 10.5802/aif.2140 %G en %F AIF_2005__55_6_1765_0
Haine, Luc. The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1765-1788. doi : 10.5802/aif.2140. http://archive.numdam.org/articles/10.5802/aif.2140/
[1] On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys., Volume 61 (1978), pp. 1-30 | DOI | MR | Zbl
[2] Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. (1985) no. 319 | MR | Zbl
[3] ``Huygens' principle and the bispectral problem, The Bispectral Problem (CRM Proc. Lecture Notes), Volume 14 (1998), pp. 11-30 | Zbl
[4] Hadamard's problem and Coxeter groups: new examples of Huygens' equations, Funktsional. Anal. i Prilozhen. 28 (1) (1994), 3-15 (Russian); English transl., Funct. Anal. Appl., Volume 28 (1994) no. 1, pp. 3-12 | MR | Zbl
[5] Expansions in Eigenfunctions of Selfadjoint Operators (Transl. Math. Monographs), Volume 17 (1968) | Zbl
[6] Integrability and Huygens' principle on symmetric spaces, Commun. Math. Phys., Volume 178 (1996), pp. 311-338 | DOI | MR | Zbl
[7] Multidimensional Baker-Akhiezer functions and Huygens' principle, Commun. Math. Phys., Volume 206 (1999), pp. 533-566 | DOI | MR | Zbl
[8] Toda flows with infinitely many variables, J. Funct. Anal., Volume 64 (1985), pp. 358-402 | DOI | MR | Zbl
[9] Inverse scattering on the line, Comm. Pure Appl. Math., Volume 32 (1979), pp. 121-251 | DOI | MR | Zbl
[10] Formulas for q-spherical functions using inverse scattering theory of reflectionless Jacobi operators, Commun. Math. Phys., Volume 210 (2000), pp. 335-369 | DOI | MR | Zbl
[11] Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and abelian varieties, Uspekhi Mat. Nauk. 31 (1) (1976), 55-136 (Russian); English transl., Russ. Math. Surveys, Volume 31 (1976) no. 1, pp. 59-146 | MR | Zbl
[12] Differential equations in the spectral parameter, Commun. Math. Phys., Volume 103 (1986), pp. 177-240 | DOI | MR | Zbl
[13] On the Toda lattice II - Inverse scattering solution, Progr. Theor. Phys., Volume 51 (1974) no. 3, pp. 703-716 | MR | Zbl
[14] Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications, 35, Cambridge University Press, 1990 | MR | Zbl
[15] Some bispectral musings, The Bispectral Problem (CRM Proc. Lecture Notes), Volume 14 (1998), pp. 31-45 | Zbl
[16] The bispectral problem: an overview, Special Functions 2000: Current Perspective and Future Directions (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.) (2001), pp. 129-140 | Zbl
[17] Some functions that generalize the Askey-Wilson polynomials, Commun. Math. Phys., Volume 184 (1997), pp. 173-202 | DOI | MR | Zbl
[18] Associated polynomials, spectral matrices and the bispectral problem, Meth. Appl. Anal., Volume 6 (1999) no. 2, pp. 209-224 | MR | Zbl
[19] Heat kernel expansions on the integers, Math. Phys. Anal. Geom., Volume 5 (2002), pp. 183-200 | DOI | MR | Zbl
[20] The q-hypergeometric equation, Askey-Wilson type solitons andrational curves with singularities, The Kowalevski Property (CRM Proc. Lecture Notes), Volume 32 (2002), pp. 69-91 | Zbl
[21] Commutative rings of difference operators and an adelic flag manifold, Internat. Math. Res. Notices, Volume 6 (2000), pp. 281-323 | MR | Zbl
[22] A rational analogue of the Krall polynomials, J. Phys. A: Math. Gen., Volume 34 (2001), pp. 2445-2457 | DOI | MR | Zbl
[23] Askey-Wilson type functions, with bound states The Ramanujan Journal (in press) | Zbl
[24] The basic Bessel functions and polynomials, SIAM J. Math. Anal., Volume 12 (1981), pp. 454-468 | DOI | MR | Zbl
[25] The application of basic numbers to Bessel's and Legendre's functions, Proc. London Math. Soc., Volume 2 (1905) no. 2, pp. 192-220 | JFM
[26] Integration in Function Spaces and Some of Its Applications, Accademia Nazionale Dei Lincei Scuola Normale Superiore, Lezion, Pisa, 1980 | MR | Zbl
[27] Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), 215-216 (Russian); English transl., Volume 33 (1978), pp. 255-256 | MR | Zbl
[28] A method of generating classes of Huygens' operators, J. Math. Mech., Volume 17 (1967) no. 5, pp. 461-472 | MR | Zbl
[29] A solution of Hadamard's problem for a restricted class of operators, Proc. Amer. Math. Soc., Volume 19 (1968), pp. 981-988 | MR | Zbl
[30] The spectrum of Hill's equation, Inventiones Math., Volume 30 (1975), pp. 217-274 | DOI | MR | Zbl
[31] The spectrum of difference operators and algebraic curves, Acta Math., Volume 143 (1979), pp. 93-154 | DOI | MR | Zbl
[32] Bispectral rings of difference operators, Russ. Math. Surveys, Volume 54 (1999), pp. 644-645 | DOI | MR | Zbl
[33] An explicit expression for the Korteweg-de Vries hierarchy, Z. Anal. Anwendungen, Volume 7 (1988), pp. 203-214 | MR | Zbl
[34] Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math., Volume 61 (1985), pp. 5-65 | DOI | Numdam | MR | Zbl
[35] Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences, 20, Springer, Berlin, Heidelberg, New-York,, 1981 | MR | Zbl
[36] Bispectral commutative ordinary differential operators, J. Reine Angew. Math., Volume 442 (1993), pp. 177-204 | MR | Zbl
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