A New Class of Exactly Solvable Models within the Schrödinger Equation with Position Dependent Mass

Abstract

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Position-dependent mass Dirac equation and local Fermi velocity

“…The scenario of PDM [36] has been widely studied in the literature. These include setting up of an extended scheme of Schrödinger equation to generate the associated class of potentials [37] and examining the consequences of a deformed shape invariance condition [38], a general strategy to tackle solvable potentials [39], exploring point canonical transformation [40], seeking new types of exact solutions for an effective mass system [41,42], investigating on the sphere and hyperbolic plane [43], looking for invariants and spectrum generating algebras [44], analyzing consistency of indefinite effective mass [45], and obtaining analytical results for PDM problems [46]. Interest in PDM systems was triggered by the physical problems pertaining to compositionally graded crystals [47], quantum dots [48], liquid crystals [49] etc.…”

Generalized Uncertainty principle and momentum-dependent effective mass Schrodinger equation

“…First, a few words about the coordinate-dependent mass picture in the configuration space that has been studied widely in the literature [17][18][19][20][21][22][23][24][25][26][27]. The interest in such systems stem essentially from the physical problems underlying compositionally graded crystals [28], quantum dots [29], liquid crystals [30] etc.…”

So (2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation

“…In a PDM setting, one has to confront an extended form of the Schrödinger equation that depends on a wide range of effective potentials containing different choices of ambiguity parameters [17][18][19]. The presence of such ambiguity parameters has indeed opened up many pathways for exploration (see, for example, [20][21][22][23][24][25]). In particular, Quesne used extensively the point canonical transformation (PCT) to analyze different variants of systems endowed with PDM [26][27][28][29].…”