$$\mathcal{P}\mathcal{T}$$ -symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach

Abstract

To lowest order of perturbation theory we show that an equivalence can be established between a PT -symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian h. An important feature of h is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical quantities. : 03.65.-w The interplay between pseudo-Hermitian PT -symmetric Hamiltonians and their equivalent Hermitian representation is currently a matter of active research [1][2][3][4][5]. While several case studies already exist in the literature, Mostafazadeh, in particular, has also considered [1] the related issue of transition to the classical limit. Quite significantly, he has observed that the underlying classical Hamiltonian for the P T -symmetric cubic anharmonic oscillator (PTCAO), in presence of a harmonic term, reveals the characteristics of a point particle that is endowed with a position-dependent mass (PDM) interacting with a quartic anharmonic field. PACSMotivated by Mostafazadeh's work, we recently developed [6] an algorithm that affords a categorization of a whole class of quantal PDM Hamiltonians in terms of perturbatively equivalent PT -symmetric counterparts with configuration space R. The PTCAO is one such model, which was shown to have a Hermitian PDM partner Hamiltonian.Keeping PT symmetry intact, the PTCAO scheme can obviously be extended to include the one-parameter family of potentials x 2 (ix) δ , δ real. It is curious to note that δ = 2 implies quartic anharmonicity but with a wrong sign (see e.g. [7]). However, as Bender and Boettcher argued [8] (see also [9]), such a feature facilitates its quasi-exact solvability.In this paper, we take up the study of a generalized scheme [10, 11] of P Tsymmetric quartic anharmonic oscillator (PTQAO) in the spirit of the perturbative analysis of [6] and examine when it should map to a Hermitian PDM Hamiltonian. We also estimate the fourth-order contribution to it and write down the physical

Generalized Laguerre polynomials with position-dependent effective mass visualized via Wigner’s distribution functions

“…Its solutions has been found to be very useful in describing, physically, the properties of the quantum dynamics of electrons in condensed matter physics as well as related fields of physics [13,14,15,16]. In mathematical physics, they have found to be interesting in point of view of coherent states [17,18,19,20] and PT -symmetry [21,22,23]. A lot of methods and approaches, including the factorization method [24], supersymmetry of quantum mechanics and the related shape-invariant potentials [25], Lie algebra [26,27,28], path integral [29] and operators techniques [30], have been applied to the system with PDEM to obtain algebraically the exact solutions.…”

A General type of Liénard Second Order Differential Equation: Classical and Quantum Mechanical Study

“…(12) . Further, we find an extensive study on the position dependence of mass pertaining to various aspects have been carried out by several authors (von Roos, 1983;Quesne and Tkachuk, 2004;Koç and Tütüncüler, 2003;Dutra et al, 2003;Bagchi et al, 2006;Ganguly et al, 2006;Ganguly and Nieto, 2007;Lévai and Özer, 2010;Killingbeck, 2011;Mazharimousavi, 2012;Yahiaoui and Bentaiba, 2012;Mustafa, 2015;Rajbongshi and Singh, 2015). In the present case, we study the discrete nature of spectra and its stability for the newly generated Hamiltonian (Eqn.…”

Position-dependent mass, finite-gap systems, and supersymmetry

“…Quantum mechanical systems with PDM were employed recently in the context of integrable models [12]. They turn out to be interesting from the point of view of supersymmetric quantum mechanics [13,14,15,16,17], coherent states [18,19,20], and PT-symmetry [21,22,23]. Besides the onedimensional quantum systems with PDM, their multi-dimensional generalizations are considered in the literature [24,25,26,27,28,29], particularly, in the context of superintegrable systems [30,31].…”