  Following the recent work [13] fulfilled in the discrete case, we pro- vide
in this paper new intertwining relations for semigroups of one-dimensional
diffusions. Various applications of these results are investigated, among them
the famous variational formula of the spectral gap derived by Chen and Wang
[15] together with a new criterion ensuring that the logarithmic Sobolev
inequality holds. We complete this work by revisiting some classical examples,
for which new estimates on the optimal constants are derived.
