Dirac Oscillators and Quasi-Exactly Solvable Operators

Abstract

The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators. 0

Quasi-exact solvability of Dirac equation with Lorentz scalar potential

“…Our classification and construction are based on the factorizability, or supersymmetric structure, of the system, with sl (2) as the underlying symmetry. There are QES systems which are not factorizable, such as those considered in [9,13,16,17], and systems which are not related to the Lie algebra sl (2), such as that discussed in [10]. It is an interesting and challenging task to develop new methods to classify and construct QES potentials in multi-component wave equations with any Lie algebraic structures.…”

“…The Pauli equation and the Dirac equation coupled minimally to a stationary vector potential were also shown to be QES [11]. More recently, added to the list are the Dirac-Pauli equation coupled non-minimally to external electric fields [12], and Dirac oscillator with Coulomb interaction supplemented by a linear radial Lorentz scalar potential [13].…”

PHASE SHIFT ANALYSIS OF PT-SYMMETRIC NONHERMITIAN EXTENSION OF AN-1 CALOGERO MODEL WITHOUT CONFINING INTERACTION

“…As concrete examples of P T symmetric quantum mechanics, the Hamiltonians of only one particle in one space dimension have mostly been considered in the literature so far. However, nonhermitian but PT invariant extension of some exactly solvable many particle quantum mechanical system in one space dimension have also been considered recently [11,12,13,14]. The PT transformation for such N-particle system can be written as…”

$\mathcal {P}\mathcal {T}$ -Symmetric Klein-Gordon Oscillator

“…The study of relativistic oscillator is of special interest in particle physics. Following the study of Dirac oscillator first proposed by Moshinsky and Szczepaniak [11], its physical applications and extensions to other case have attracted a lot of attention and been studied intensively by various authors [12][13][14][15][16][17][18][19][20][21][22][23][24]. …”