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

Abstract

Parity-time (PT ) symmetric Klein-Gordon oscillator is presented using PTsymmetric minimal substitution. It is shown that wave equation is exactly solvable, and energy spectrum is the same as that of Hermitian Klein-Gordon oscillator presented by Bruce and Minning. Landau problem of PT -symmetric Klein-Gordon oscillator is discussed.Keywords Klein-Gordon oscillator · PT symmetry · Landau problem Although Hermitian Hamiltonian holds mainstream for quantum mechanics and quantum field theory, the study of non-Hermitian Hamiltonian has attracted a great deal of attention. This interest is triggered by the development of the studies of PT -symmetric Hamiltonian. In a fundamental paper [1] Bender and Boettcher found that it is possible to see that the energy eigenvalues of non-Hermitian Hamiltonians such as H = p 2 + x 2 (ix) ( ≥ 0) are all real as long as they preserve PT symmetry. Then, Dorey et al. provided a rigorous proof of spectral positivity [2,3]. In [4,5], it was shown that the time-evolution operator for the PT -symmetric Hamiltonian is unitary. A large number of PT -symmetric models have been studied, including PT -symmetric quantum mechanics [6], PT -symmetric quantum electrodynamics [7], PT -symmetric quantum field theory [8] as well as optical PT -symmetric structures [9]. For a comprehensive review of the basic ideas and techniques responsible for the recent developments in non-Hermitian Hamiltonian, see Ref. [10].Harmonic oscillator is one of the most useful and well studied system. In the nonrelativistic limit, the positive energy states of relativistic oscillator reduces to the spectrum of non-relativistic harmonic 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].

A relativistic quantum oscillator subject to a Coulomb-type potential induced by effects of the violation of the Lorentz symmetry

“…It was proposed by Bruce and Minning [45] in analogy with the Dirac oscillator [49] and it has attracted interests in studies of noncommutative space [50,51], in noncommutative phase space [52], in Kaluza-Klein theories [53] and in PT -symmetric Hamiltonian [54]. The relativistic oscillator coupling proposed by Bruce and…”

Effects of a linear central potential induced by the Lorentz symmetry violation on the Klein–Gordon oscillator

“…Inspired by the relativistic oscillator model for the spin-1 2 fermionic field known as the Dirac oscillator [27,28], Bruce and Minning have proposed a relativistic oscillator model for the scalar field which it was known in the literature as the Klein-Gordon oscillator [29] that, in the nonrelativistic limit, is reduced to the oscillator described by the Schoröndinger equation [30]. The Klein-Gordon oscillator has been studied by a PT -symmetric Hamiltonian [31], in noncommutative space [32,33], in spacetime with cosmic string [34], in a spacetime with torsion [35], in a Kaluza-Klein theory [36], with noninertial effects [37] and under effects of linear and Coulomb-type central potentials [38][39][40].…”

On the Klein–Gordon oscillator subject to a Coulomb-type potential

“…where m is the rest mass of the scalar particle, ω is the angular frequency of the Klein-Gordon oscillator, ρ = x 2 + y 2 andρ is a unit vector in the radial direction. In recent years, the Klein-Gordon oscillator has been investigated in noncommutative space [16,17], in noncommutative phase space [18] and in PT -symmetric Hamiltonian [19].…”