  Dealing with one-dimensional diffusion operators, we obtain upper and lower
variational formulae on the eigenvalues given by the max-min principle,
generalizing the celebrated result of Chen and Wang on the spectral gap. Our
inequalities reveal to be sharp at least when the eigenvalues considered belong
to the discrete spectrum of the operator, since in this case both lower and
upper bounds coincide and involve the associated eigenfunctions. Based on the
intertwinings between diffusion operators and some convenient gradients with
weights, our approach also allows to estimate the gap between the two first
positive eigenvalues when the spectral gap belongs to the discrete spectrum.
