Symbolic estimation of distances between eigenvalues of Hermitian operator and unitary orbits classification

TL;DR

This paper introduces symbolic methods using trace expressions to analyze eigenvalue distances, polynomial factorization, and classify unitary orbits of Hermitian operators with rational functions of polynomial coefficients.

Contribution

It presents a novel symbolic approach for eigenvalue analysis and orbit classification based solely on characteristic polynomial coefficients.

Findings

Factorization of characteristic polynomial into multiplicity-based factors.

Accurate estimation of minimal eigenvalue distances using rational functions.

Classification of Hermitian operator unitary orbits.

Abstract

We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of characteristic polynomial that collects all eigenvalues that have the same multiplicity into one polynomial factor with coefficients rationally expressed through coefficients of original characteristic polynomial. Finding with any accuracy a minimal distance between eigenvalues, as well as a minimal and a maximal eigenvalues of the Hermitian operator, applying only rational functions of corresponding coefficients of the characteristic polynomial. Estimation of related rate of convergence. Classification of unitary orbits of Hermitian operators.