Two anyons with Coulomb interaction in a magnetic field

On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential

“…From Refs. [40][41][42], the behaviour of the possible solutions to Eq. (10) at r → 0 and r → ∞ allows us to write the function R (r) in terms of an unknown function H (r) as follows:…”

“…(14). Thereby, bound state solutions can be achieved by imposing that the power series expansion (14) or the biconfluent Heun series becomes a polynomial of degree n. In this way, we guarantee that R (r) behaves as r |γ| at the origin and vanishes at r → ∞ [41,42]. Through the recurrence relation (15), we can see that the power series expansion (14) becomes a polynomial of degree n by imposing two conditions …”

“…Furthermore, the linear scalar potential has attracted a great interest in atomic and molecular physics as pointed out in Refs. [56][57][58][59][60][61] and in several discussions of relativistic quantum mechanics [39][40][41][42][62][63][64][65][66][67][68].…”

Quantum Effects on an Atom with a Magnetic Quadrupole Moment in a Region with a Time-Dependent Magnetic Field

“…It is worth pointing out that in the context of condensed matter physics, scalar potentials proportional to the inverse of the radial distance have been studied with 1-dimensional systems [45][46][47][48][49], molecules [50][51][52], pseudo-harmonic interactions [53,54], position-dependent mass systems [55][56][57], the Kratzer potential [58][59][60]. Other contexts are the propagation of gravitational waves [61], quark models [62], atoms with magnetic quadrupole moment [34] and relativistic quantum mechanics [63][64][65][66][67].…”

“…Observe that the asymptotic behaviour is determined for r → 0 and r → ∞, then, we can write the function R (r) in terms of an unknown function G (r) as follows [34,63,64,68]:…”