Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations

TL;DR

This paper explores the connection between anharmonic oscillators, thermodynamic Bethe ansatz (TBA) systems, and nonlinear integral equations, providing new integral representations and conjectures linking spectral determinants to integrable models.

Contribution

It introduces an alternative integral expression for spectral determinants of anharmonic oscillators and conjectures a general relationship with TBA systems for various potentials.

Findings

Derived integral expressions for spectral determinants.

Conjectured links between anharmonic oscillators and TBA systems.

Mapped spectral determinants to solutions of nonlinear integral equations.

Abstract

The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for $D(E)$. Generalising this result, we conjecture a relationship between the $x^{2M}$ anharmonic oscillators and the $A_{2M-1}$ TBA systems. Finally, spectral determinants for general $|x|^{\alpha}$ potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models.