Algebraic approach to the inverse spectral problem for rational matrices

Marco Bertola

Paper

Algebraic approach to the inverse spectral problem for rational matrices

Abstract

We consider the problem of reconstruction of an

matrix with coefficients depending rationally on

from the data of: (a) its characteristic polynomial and (b) a line bundle of degree

, with

the geometric genus of the spectral curve, represented by a choice of

points forming a (non-positive) divisor of the given degree. We thus provide a reconstruction formula that does not involve transcendental functions; this includes formulas for the spectral projectors and for the change of line bundle, thus integrating the isospectral flows. The formula is a single residue formula which depends rationally on the coordinates of the points involved, the coefficients of the spectral curve, and the position of the finite poles of

. We also discuss the canonical bi-differential associated with the Lax matrix and its relationship with other bi-differentials that appear in Topological Recursion and integrable systems.