Isospectral flows for the inhomogeneous string density problem

TL;DR

This paper develops isospectral flows for the inhomogeneous string density problem, providing a framework for understanding spectral invariants and explicit solutions using advanced mathematical tools.

Contribution

It introduces a new class of rational isospectral flows for the string density problem, linking them to polynomial flows and providing explicit Hamiltonian solutions.

Findings

Certain rational flows generate all polynomial-based flows in the limit

Explicit solutions are obtained via Stieltjes continued fractions and Hankel determinants

The flows are proven to be Hamiltonian systems

Abstract

We derive isospectral flows of the mass density in the string boundary value problem corresponding to general boundary conditions. In particular, we show that certain class of rational flows produces in a suitable limit all flows generated by polynomials in negative powers of the spectral parameter. We illustrate the theory with concrete examples of isospectral flows of discrete mass densities which we prove to be Hamiltonian and for which we provide explicit solutions of equations of motion in terms of Stieltjes continued fractions and Hankel determinants.