Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations

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). Generalizing this result, we conjecture a relationship between the x(2M) anharmonic oscillators and the A(2M-1) thermodynamic Bethe ansatz 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.