Algebraic approach to the inverse spectral problem for rational matrices

TL;DR

This paper presents a purely algebraic method for reconstructing rational matrices from spectral data, avoiding transcendental functions, and provides explicit formulas for spectral projectors and isospectral flows.

Contribution

It introduces a rational residue formula for inverse spectral problems of rational matrices, linking spectral data with algebraic reconstruction without transcendental functions.

Findings

Provides explicit residue formulas for matrix reconstruction.

Connects spectral data with algebraic geometry via line bundles.

Integrates isospectral flows through algebraic formulas.

Abstract

We consider the problem of reconstruction of an $n\times n$ matrix with coefficients depending rationally on $x\in \mathbb P^1$ from the data of: (a) its characteristic polynomial and (b) a line bundle of degree $g+n-1$, with $g$ the geometric genus of the spectral curve, represented by a choice of $g+n+1$ 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 $L$. 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.