The bound state Aharonov-Bohm effect around a cosmic string revisited

When a quantum particle is confined between two impenetrable concentric cylindrical walls, threaded by a magnetic flux tube along their common axis, its energy spectrum presents a dependence on the flux. This is the so-called bound state Aharonov-Bohm effect [1]. A gravitational analogue of this effect, with a cosmic string replacing the flux tube, has been studied in references [2, 3]. This is a topological effect which also appears around disclinations in elastic solids [4]. In this article we revisit the bound state gravitational Aharonov-Bohm effect from the point of view of self-adjoint extensions of the Hamiltonian, without need of a confining wall. Fermions in the presence of an Aharonov-Bohm field may have a bound state appearing from the self-adjoint extension of the Dirac Hamiltonian, as shown in [5], for a specific range of the extension parameter. As shown in the detailed analysis of Kay and Studer [6], a bound state may appear for a quantum particle moving about a cosmic string due to the self-adjoint extension of its Hamiltonian. On the other hand, the presence of walls may also lead to interesting effects related to the selfadjoint extensions as well [7]. In this article we analyze the idealized situation of a free non-relativistic quantum particle in the presence of a cosmic string and therefore, moving in the background spacetime of metric