Browsing by Author "Pozzoli, Eugenio"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Geometric Confinement and Dynamical Transmission of a Quantum Particle in Grushin Cylinder
    (SISSA, 2019) Gallone, Matteo; Michelangeli, Alessandro; Pozzoli, Eugenio
    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.
  • Thumbnail Image
    Item
    On Geometric Quantum Confinement in Grushin-Like Manifolds
    (2018-09) Gallone, Matteo; Michelangeli, Alessandro; Pozzoli, Eugenio; Mathematics
    We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.