The spectral determinant D(E) of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the A₃-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| potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models.