Embedding the Grushin Cylinder in ${\mathbf{R}}^3$and Schroedinger evolution

We consider the evolution of a free quantum particle on the Grushin cylinder, under different type of quantizations. In particular we are interested to understand if the particle can cross the singular set, i.e., the set where the structure is not Riemannian. We consider intrinsic and extrinsic quantizations, where the latter are obtained by embedding the Grushin structure isometrically in ${\mathbf{R}}^3$ (with singularities). As a byproduct we provide formulas to embed the Grushin cylinder in ${\mathbf{R}}^{\mathbf{3}}$ that could be useful for other purposes. Such formulas are not global, but permit to study the embedding arbitrarily close to the singular set. We extend these results to the case of $\alpha$-Grushin cylinders.