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Item Discrete spectra for critical Dirac-Coulomb Hamiltonians(2017) Gallone, Matteo; Michelangeli, Alessandro; MathematicsShow more 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.Show more Item Geometric Confinement and Dynamical Transmission of a Quantum Particle in Grushin Cylinder(SISSA, 2019) Gallone, Matteo; Michelangeli, Alessandro; Pozzoli, EugenioShow more 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.Show more Item Hydrogenoid Spectra with Central Perturbations(2018-08) Gallone, Matteo; Michelangeli, Alessandro; MathematicsShow more 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.Show more Item Krein-Visik-Birman self-adjoint extension theory revisited(2017) Gallone, Matteo; Michelangeli, Alessandro; Ottolini, Andrea; MathematicsShow more 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.Show more Item On Geometric Quantum Confinement in Grushin-Like Manifolds(2018-09) Gallone, Matteo; Michelangeli, Alessandro; Pozzoli, Eugenio; MathematicsShow more 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.Show more Item Self-Adjoint Extensions of Dirac Operator with Coulomb Potential(SISSA, 2017-02) Gallone, MatteoShow more 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.Show more Item Self-adjoint extensions with Friedrichs lower bound(SISSA, 2020) Gallone, Matteo; Michelangeli, AlessandroShow more We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.Show more Item Self-Adjoint Extensions with Friedrichs Lower Bound(2020) Gallone, Matteo; Michelangeli, Alessandro; mathematicsShow more We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussedShow more Item Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei(2017-06) Gallone, Matteo; Michelangeli, Alessandro; MathematicsShow more 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.Show more Item UNIVERSALITY IN THE 2D QUASI-PERIODIC ISING MODEL AND HARRIS-LUCK IRRELEVANCE(2023-04-04) Gallone, Matteo; Mastropietro, Vieri; mathematicsShow more We prove that in the 2d Ising Model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case, that is the critical exponents are identical and no logarithmic corrections are present. The result establishes the validity of the prediction based on the Harris-Luck criterion and it provides the first rigorous proof of universality in the Ising model in presence of quasi-periodic disorder. The proof combines Renormalization Group approaches with direct methods used to deal with small divisors in KAM theory.Show more