Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
(1999) Communications in Mathematical Physics — Vol. 207, n° 3, p. 589-620 (1999)
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- Abstract
- Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t = (t(1), t(2), ...), leads to the standard Toda lattice and tau-functions, expressed as hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated tau-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero-Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m + 1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann-Hilbert problem. We show the Riemann-Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower-times upper-triangular matrix.
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Adler, M., & Van Moerbeke, P. (1999). Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems. Communications in Mathematical Physics, 207(3), 589-620. https://doi.org/10.1007/s002200050738 (Original work published 1999)