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

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.