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## 1 Introduction 

Hamiltonians for particles endowed with position-dependent-mass (PDM) (i.e., M ( x ) = m$_{◦}$m ( x )) have attracted much research attention over the last few decades [1-29]. Such attention was inspired not only by the feasible applica- bility of PDM-settings in the study of various physical problems (e.g., many- body problem, semiconductors, quantum dots, quantum liquids, etc.) but also by the mathematical challenge associated with the ordering ambiguity in the PDM van Roos Hamiltonian. The non-commutativity between the momentum operator (with ℏ = m$_{◦}$ = 1 units to be used through out) ˆ p$_{x}$ = − i∂$_{x}$ and the position-dependent-mass results in an ordering ambiguity in the non-uniqueness representation of the kinetic energy operator 

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where α , β , and γ are called the van Roos ordering ambiguity parameters sat- isfying the van Roos constraint α + β + γ = − 1 [cf., e.g., 25-29]. 

In the literature, there exist several suggestions for the van Roos order- ing ambiguity parameters. Amongst, the Gora’s and Williams’ ( β = γ = 0 , α = − 1), Ben Daniel’s and Duke’s ( α = γ = 0 , β = − 1), Zhu’s and Kroemer’s ( α = γ = − 1 / 2 , β = 0) , Li’s and Kuhn’s ( β = γ = − 1 / 2 , α = 0), and the very recent Mustafa’s and Mazharimousavi’s ( α = γ = − 1 / 4 , β = − 1 / 2) (cf. e. g., [10, 29] for more details on this issue). It has been observed (cf., e. g., [29,40]) that the physical and/or mathematical admissibility of a given ambigu- ity parameters set very well depends not only on the continuity conditions at the abrupt heterojunction boundaries but also on the position-dependent-mass form and/or potential form. The general consensus is that there is no unique neither there is a universal choice for these ambiguity parameters, therefore. 

## 2 

