Self-Adjoint Extensions of Dirac Operator with Coulomb Potential

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 $V(\vec x)=\phi(\vec x) I_4$. 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.Comment: 17 page

Discrete spectra for critical Dirac-Coulomb Hamiltonians

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.Comment: 24 pages, 3 figures. Version published on Journal of Mathematical Physics (2018

On Geometric Quantum Confinement in Grushin-type Manifolds

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.Comment: 16 pages, 2 figure

Superselection Structure of Massive Quantum Field Theories in 1+1 Dimensions

We show that a large class of massive quantum field theories in 1+1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.Comment: latex2e, 21 pages. Final version, to appear in Rev. Math. Phys. Some improvements of the presentation, but no essential change

Self-adjointness of Quantum Hamiltonians with Symmetries

This thesis discusses the general problem of the self-adjoint realisation of formal Hamiltonians with a focus on a number of quantum mechanical models of actual relevance in the current literature, which display certain symmetries. In the first part we analyse the general extension theory of (possibly unbounded) linear operators on Hilbert space, and in particular we revisit the Kre\u{i}n-Vi{s}ik-Birman theory that we are going to use in the applications. We also discuss the interplay between extension theory and presence of discrete symmetries, which is the framework of the present work. The second part of the thesis contains the study of three explicit quantum models, two that are well-known since long and a more modern one, each of which is receiving a considerable amount of attention in the recent literature as far as the identification and the classification of the extensions is concerned. First we characterise all self-adjoint extensions of the Hydrogen Hamiltonian with point-like interaction in the origin and of the Dirac-Coulomb operators. For these two operators we also provide an explicit formula for the eigenvalues of every self-adjoint extension and a characterisation of the domain of respective operators in term of standard functional spaces. Then we investigate the problem of geometric quantum confinement for a particle constrained on a Grushin-type plane: this yields the analysis of the essential self-adjointness for the Laplace-Beltrami operator on a family of Riemannian manifolds

Prethermalization and conservation laws in quasi-periodically-driven quantum systems

We study conservation laws of a general class of quantum many-body systems subjected to an external time dependent quasi-periodic driving. We show that, when the frequency of the driving is large enough or the strength of the driving is small enough, the system exhibits a prethermal state for stretched exponentially long times in the perturbative parameter. Moreover, we prove the quasi-conservation of the constants of motion of the unperturbed Hamiltonian and we analyze their physical meaning in examples of relevance to condensed matter and statistical physics.Comment: 41 pages; 3 figures; revised version. Several comments adde

Self-adjoint extensions with Friedrichs lower bound

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