Copyright © The Author 2007. Published by Oxford University Press.
Effective Inverse Spectral Problem for Rational Lax Matrices and Applications
1 Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. Centre ville, Montréal, Québec, Canada H3C 3J7,
2 Department of Mathematics and Statistics, Concordia University, 1400 Sherbrooke W., Montréal (QC), H4B 1R6,
3 Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556-4618
Correspondence: Correspondence to be sent to: M. Bertola, Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. Centre ville, Montréal, Québec, Canada H3C 3J7. e-mail: bertola{at}mathstat.concordia.ca
We reconstruct a rational Lax matrix of size R + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy-like kernel (a bi-differential of bi-weight (1 – 
, )) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights , 1 – respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of "elementary twists", we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices.
Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite–band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance of formulæ obtained to the study of the large degree asymptotics for these polynomials is indicated.