Geometric Confinement and Dynamical Transmission of a Quantum Particle in Grushin Cylinder
We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally deﬁned on an inﬁnite cylinder equipped with an incom-plete Riemannian metric of Grushin type, in the non-trivial class of metrics yielding an inﬁnite deﬁciency index. Such realisations are naturally interpreted as Hamiltonians governing the geometric conﬁnement of a Schr¨odinger quan-tum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterise all physically meaningful exten-sions 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.
Geometry and Mathematical Physics