Darboux transforms on Band Matrices, Weights and associated Polynomials
Adler, Mark ; van Moerbeke, Pierre
arXiv, 0010048 / Harvested from arXiv
Text on arXiv
Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have shown that $m$-periodic sequences of weights lead to "moments", polynomials defined by determinants of matrices involving these moments and $2m+1$-step relations between them, thus leading to $2m+1$-band matrices $L$. Given a Darboux transformations on $L$, which effect does it have on the $m$-periodic sequence of weights and on the associated polynomials ? These questions will receive a precise answer in this paper. The methods are based on introducing time parameters in the weights, making the band matrix $L$ evolve according to the so-called discrete KP hierarchy. Darboux transformations on that $L$ translate into vertex operators acting on the $\tau$-function.
Publié le : 2000-10-27
Classification:  Nonlinear Sciences - Exactly Solvable and Integrable Systems,  Mathematical Physics,  Mathematics - Classical Analysis and ODEs 
@article{0010048,
     author = {Adler, Mark and van Moerbeke, Pierre},
     title = {Darboux transforms on Band Matrices, Weights and associated Polynomials},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0010048}
}

Adler, Mark; van Moerbeke, Pierre. Darboux transforms on Band Matrices, Weights and associated Polynomials. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0010048/