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.