Self-Adjoint Extension Approach for Singular Hamiltonians in (2 + 1) Dimensions

Abstract

In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions.In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state. However, a study based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.

Effects of rotation and Coulomb type potential on the spin-1/2 Aharonov-Bohm problem

“…[66] (see also Refs. [55]). This boundary condition is a mathematical limit that allows divergent solutions for the Hamiltonian H 0 at isolated regions, provided they remain square-integrable.…”

“…Another topic of central importance in this context refers to the description of the AB effect by taking into account the electron spin degree of freedom [54]. As we shall see in more detail below, this problem demands particular mathematical tools for adequate treatment [55]. When a magnetic field is present, we know that the spin degree of freedom is responsible for lifting the degeneracy of the energy levels of a particle due to the Zeeman interaction.…”

Observables in Quantum Mechanics and the Importance of Self-Adjointness

“…Self-adjointness and the theory of SA extensions are known to play important roles in a variety of physical contexts, including systems with a confined particle [21][22][23], Aharonov-Bohm effect [24][25][26][27], graphene [28], two and three dimensional delta function potentials [29], heavy atoms [30][31][32], singular potentials [33,34], Calogero models [35,36], anyons [37,38], anomalies [39][40][41], ζ-function renormalization [42], scattering theory [43], particle statistics [44], black holes [45][46][47][48][49], integrable system [50,51], Klein-Gordon equation [52], renormalons in QM [53], quasinormal modes [54], supersymmetric QM [55] and toy models for strings [56], spectral triple [57], noncommutative field theories [58][59][60], resolving the spacetime singularities [61][62][63][64][65] and even pl...…”

Relativistic Quantum Motion of an Electron in Spinning Cosmic String Spacetime in the Presence of Uniform Magnetic Field and Aharonov-Bohm Potential

“…[36], for example, it was considered the interaction of a point charge with a magnetic field in a spacetime with a distortion. References [37][38][39][40][41][42][43][44] are examples of studies dealing with Landau levels and the Aharonov-Bohm effect in the presence of topological defects. Reference [45] deals with a quantum ring in graphene with a topological defect and a magnetic field.…”