Effective Inverse Spectral Problem for Rational Lax Matrices and Applications

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 − ν, ν)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν, 1 − ν 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.