Gallone, MatteoMichelangeli, AlessandroPozzoli, Eugenio2018-09-192018-09-192018-09https://openscience.sissa.it/handle/1963/3532216 pagesWe 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.enGeometric quantum confinementGrushin manifoldgeodesically (in)complete Riemannian manifoldLaplace-Beltrami operatoralmost-Riemannian structureself-adjoint operators in Hilbert spaceWeyl's limit-point limit-circle criterionconstant-fibre direct integral.Preprint