Generalized Weyl algebras, as defined by V. V. Bavula [St. Petersburgh
Math. J., 1993], are a family of algebras containing both some classical
objects (enveloping algebras and their prime quotients, Weyl algebras,
invariant sub-algebras,...) and their quantum analogues. These algebras
are generated by two generators over a k-algebra R, with relations given
thanks to an automorphism and a central element of R.

We are interested in problems of classification for such algebras. In
this talk based on a joint work with L. Richard, we will consider
isomorphisms between generalized Weyl algebras, giving a complete answer
to this problem in the quantum case for R = k[h]. We will give
separation results too up to rational equivalence and
Morita-equivalence for these algebras.