%0 Report
%D 2018
%T On Geometric Quantum Confinement in Grushin-Like Manifolds
%A Matteo Gallone
%A Alessandro Michelangeli
%A Eugenio Pozzoli
%X 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.
%G en
%U http://preprints.sissa.it/handle/1963/35322
%1 35632
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-09-19T07:05:15Z
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%0 Report
%D 2018
%T Hydrogenoid Spectra with Central Perturbations
%A Matteo Gallone
%A Alessandro Michelangeli
%X Through 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.
%G en
%U http://preprints.sissa.it/handle/1963/35321
%1 35631
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-08-27T06:42:30Z
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%0 Report
%D 2017
%T Discrete spectra for critical Dirac-Coulomb Hamiltonians
%A Matteo Gallone
%A Alessandro Michelangeli
%X The 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.
%G en
%U http://preprints.sissa.it/handle/1963/35300
%1 35606
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-11-06T14:27:38Z
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%0 Report
%D 2017
%T Krein-Visik-Birman self-adjoint extension theory revisited
%A Matteo Gallone
%A Alessandro Michelangeli
%A Andrea Ottolini
%X The 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.
%G en
%U http://preprints.sissa.it/handle/1963/35286
%1 35591
%2 Mathematics
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-05-31T08:18:47Z
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%0 Report
%D 2017
%T Self-Adjoint Extensions of Dirac Operator with Coulomb Potential
%A Matteo Gallone
%X In 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.
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35273
%1 35579
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-02-17T06:59:43Z
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%0 Report
%D 2017
%T Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei
%A Matteo Gallone
%A Alessandro Michelangeli
%X We 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.
%G en
%U http://preprints.sissa.it/handle/1963/35287
%1 35592
%2 Mathematics
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-06-05T06:42:39Z
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