@article {2018,
	title = {On Geometric Quantum Confinement in Grushin-Like Manifolds},
	number = {SISSA;36/2018/MATE},
	year = {2018},
	note = {16 pages},
	abstract = {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{\textquoteright}s analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.},
	url = {http://preprints.sissa.it/handle/1963/35322},
	author = {Matteo Gallone and Alessandro Michelangeli and Eugenio Pozzoli}
}
@article {2018,
	title = {Hydrogenoid Spectra with Central Perturbations},
	number = {SISSA;34/2018/MATE},
	year = {2018},
	note = {Mathematics Subject Classification (2010) 34L10 . 34L15 . 34L16 . 47B15 . 47B25 . 47N20 . 81Q10 . 81Q80},
	abstract = {Through the Krein-Vi{\v s}ik-Birman extension scheme, unlike the previous
classical analysis based on von Neumann{\textquoteright}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 Sch{\"o}rdinger 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 Krein-Vi{\v s}ik-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.},
	url = {http://preprints.sissa.it/handle/1963/35321},
	author = {Matteo Gallone and Alessandro Michelangeli}
}
@article {2017,
	title = {Discrete spectra for critical Dirac-Coulomb Hamiltonians},
	number = {SISSA;44/2017/MATE},
	year = {2017},
	abstract = {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{\textquoteright}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{\textquoteright}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.},
	url = {http://preprints.sissa.it/handle/1963/35300},
	author = {Matteo Gallone and Alessandro Michelangeli}
}
@article {2017,
	title = {Krein-Visik-Birman self-adjoint extension theory revisited},
	number = {SISSA;25/2017/MATE},
	year = {2017},
	abstract = {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.},
	url = {http://preprints.sissa.it/handle/1963/35286},
	author = {Matteo Gallone and Alessandro Michelangeli and Andrea Ottolini}
}
@article {2017,
	title = {Self-Adjoint Extensions of Dirac Operator with Coulomb Potential},
	number = {SISSA;09/2017/MATE},
	year = {2017},
	institution = {SISSA},
	abstract = {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) = {\O}(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.},
	url = {http://urania.sissa.it/xmlui/handle/1963/35273},
	author = {Matteo Gallone}
}
@article {2017,
	title = {Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei},
	number = {SISSA;26/2017/MATE},
	year = {2017},
	abstract = {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 KreIn-Vi{\v s}ik-
Birman extension scheme, or also on Grubb{\textquoteright}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.},
	url = {http://preprints.sissa.it/handle/1963/35287},
	author = {Matteo Gallone and Alessandro Michelangeli}
}
