00800nas a2200121 4500008004100000245006300041210005900104520039700163100002000560700002900580700002100609856004800630 2018 en d00aOn Geometric Quantum Confinement in Grushin-Like Manifolds0 aGeometric Quantum Confinement in GrushinLike Manifolds3 aWe 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.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aPozzoli, Eugenio uhttp://preprints.sissa.it/handle/1963/3532201340nas a2200109 4500008004100000245005100041210005100092520099000143100002001133700002901153856004801182 2018 en d00aHydrogenoid Spectra with Central Perturbations0 aHydrogenoid Spectra with Central Perturbations3 aThrough the Kreĭn-Višik-Birman extension scheme, unlike the previous
classical analysis based on von Neumann's theory, we reproduce the construction
and classification of all self-adjoint realisations of two intimately related models:
the three-dimensional hydrogenoid-like Hamiltonians with singular perturbation
supported at the centre (the nucleus), and the Schördinger operators on the halfline
with Coulomb potentials centred at the origin. These two problems are technically
equivalent, albeit sometimes treated by their own in the the literature. Based
on such scheme, we then recover the formula to determine the eigenvalues of each
self-adjoint extension, which are corrections to the non-relativistic hydrogenoid energy
levels.We discuss in which respect the Kreĭn-Višik-Birman scheme is somehow
more natural in yielding the typical boundary condition of self-adjointness at the
centre of the perturbation and in identifying the eigenvalues of each extension.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3532101132nas a2200109 4500008004100000245006100041210006000102520076300162100002000925700002900945856004800974 2017 en d00aDiscrete spectra for critical Dirac-Coulomb Hamiltonians0 aDiscrete spectra for critical DiracCoulomb Hamiltonians3 aThe one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general
classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3530000742nas a2200121 4500008004100000245006300041210006000104520033800164100002000502700002900522700002100551856004800572 2017 en d00aKrein-Visik-Birman self-adjoint extension theory revisited0 aKreinVisikBirman selfadjoint extension theory revisited3 aThe core results of the so-called KreIn-Visik-Birman theory of
self-adjoint extensions of semi-bounded symmetric operators are reproduced,
both in their original and in a more modern formulation, within a
comprehensive discussion that includes missing details, elucidative steps,
and intermediate results of independent interest.1 aGallone, Matteo1 aMichelangeli, Alessandro1 aOttolini, Andrea uhttp://preprints.sissa.it/handle/1963/3528600941nas a2200109 4500008004100000245006900041210006800110260001000178520057200188100002000760856005100780 2017 en d00aSelf-Adjoint Extensions of Dirac Operator with Coulomb Potential0 aSelfAdjoint Extensions of Dirac Operator with Coulomb Potential bSISSA3 aIn this note we give a concise review of the present state-of-art for the
problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular
scalar potential V(x) = Ø(x)I4. We try to follow the historical and conceptual
path that leads to the present understanding of the problem and to highlight the
techniques employed and the main ideas. In the final part we outline a few major
open questions that concern the topical problem of the multiplicity of self-adjoint
realisations of the model, and which are worth addressing in the future.1 aGallone, Matteo uhttp://urania.sissa.it/xmlui/handle/1963/3527301120nas a2200109 4500008004100000245008000041210006900121520072300190100002000913700002900933856004800962 2017 en d00aSelf-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei0 aSelfadjoint realisations of the DiracCoulomb Hamiltonian for hea3 aWe derive a classification of the self-adjoint extensions of the
three-dimensional Dirac-Coulomb operator in the critical regime of the
Coulomb coupling. Our approach is solely based upon the KreĬn-Višik-
Birman extension scheme, or also on Grubb's universal classification
theory, as opposite to previous works within the standard von Neu-
mann framework. This let the boundary condition of self-adjointness
emerge, neatly and intrinsically, as a multiplicative constraint between
regular and singular part of the functions in the domain of the exten-
sion, the multiplicative constant giving also immediate information on
the invertibility property and on the resolvent and spectral gap of the
extension.1 aGallone, Matteo1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/35287