Existence and uniqueness of solutions to Y-systems and TBA equations

TL;DR

This paper proves the existence and uniqueness of solutions to Y-systems and TBA equations under certain conditions, extending previous results to more general matrices beyond integer cases.

Contribution

It establishes existence and uniqueness results for solutions of Y-systems and TBA equations with general non-negative, irreducible, diagonalisable matrices, broadening prior work.

Findings

Proved existence and uniqueness of solutions under specified conditions.

Extended results to non-integer matrices, including adjacency matrices of Dynkin diagrams.

Established positivity of solutions for constant Y-systems.

Abstract

We consider Y-system functional equations of the form $$ Y_n(x+i)Y_n(x-i)=\prod_{m=1}^N (1+Y_m(x))^{G_{nm}}$$ and the corresponding nonlinear integral equations of the Thermodynamic Bethe Ansatz. We prove an existence and uniqueness result for solutions of these equations, subject to appropriate conditions on the analytical properties of the $Y_n$, in particular the absence of zeros in a strip around the real axis. The matrix $G_{nm}$ must have non-negative real entries, and be irreducible and diagonalisable over $\mathbb{R}$ with spectral radius less than 2. This includes the adjacency matrices of finite Dynkin diagrams, but covers much more as we do not require $G_{nm}$ to be integers. Our results specialise to the constant Y-system, proving existence and uniqueness of a strictly positive solution in that case.