Quantum geometric confinement and dynamical transmission in Grushin cylinder

We classify the self-adjoint realizations of the Laplace-Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one.