  We introduce the periodic Airy-Schr\"odinger operator and we study its band
spectrum. This is an example of an explicitly solvable model with a periodic
potential which is not differentiable at its minima and maxima. We define a
semiclassical regime in which the results are stated for a fixed value of the
semiclassical parameter and are thus estimates instead of asymptotic results.
We prove that there exists a sequence of explicit constants, which are zeroes
of classical functions, giving upper bounds of the semiclassical parameter for
which the spectral bands are in the semiclassical regime. We completely
determine the behaviour of the edges of the first spectral band with respect to
the semiclassical parameter. Then, we investigate the spectral bands and gaps
situated in the range of the potential. We prove precise estimates on the
widths of these spectral bands and these spectral gaps and we determine an
upper bound on the integrated spectral density in this range. Finally, in the
semiclassical regime, we get estimates of the edges of every spectral bands and
thus of the widths of every spectral bands and spectral gaps.
