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Position-dependent-mass; Cylindrical coordinates, separability, exact solvability, and PT-symmetry

The kinetic energy operator with position-dependent-mass in cylindrical coordinates is obtained. The separability of the corresponding Schr\"odinger equation is discussed within radial cylindrical mass settings. Azimuthal symmetry is assumed and spectral signatures of various z-dependent interaction potentials (Hermitian and non-Hermitian PT-symmetric) are reported.


Introduction
The von Roos Hamiltonian for position-dependent-mass (PDM) quantum particles is known to be associated with an ordering ambiguity problem manifested by the non-unique representation of the kinetic energy operator [1]. In such an obvious profile change in the effective potential is introduced when the parametric values of the ambiguity parameters (α, β, γ) are changed (within the von Roos constraint α + β + γ = −1). Nevertheless, it is known that the continuity conditions at the heterojunction boundaries between two crystals imply α = γ (cf., e.g., ref. [2] and the related references cited therein) . This would effectively reduce the domain of the acceptable parametric values of the ambiguity parameters. In fact, the PDM Hamiltonian (1) is known to be a descriptive model for many physical problems (like but not limited to, many-body problem, electronic properties of semiconductors, etc.) . It is, moreover, a mathematically challenging and a useful model that enriches the class of exactly solvable quantum mechanical systems.
In the literature, nevertheless, one may find many suggestion on the am- Mustafa and Mazharimousavi α = γ = −1/4, β = −1/2 (cf., e.g., [2,3] and references therein). Very recently, we have studied the problem of a singular PDM particle in an infinite potential well and shown that none of the above known parametric ordering sets is admissible within the methodical proposal discussed in [3]. Consequently, the ordering ambiguity conflict does not only depend on the heterojunction boundaries and the Dutra and Almeida's [4] reliability test (cf., e.g., Ref.s [3,4] for more details). The potential and/or the form of the position dependent mass have their say in the process [3]. At the end of the day, however, the consensus is that this ambiguity is mainly attributed to the lack of the Galilean invariance (cf., e.g., Ref. [1] on the details of this issue).
In the current methodical proposal, we shall be working with the ambiguity parameters as they are without any discrimination as to which set of ordering is favorable than which. We discuss the von Roos Hamiltonian (1) using cylindrical coordinates and seek some feasible separability in section 2.
Therein, we suggest the position-dependent-mass to be only radial-dependent and azimuthal symmetrization is sought through the assumption that Of course, this constitutes only a one feasible separability of the system (other separability options may occur as well), as justified in section 2. In section 3, within the radial cylindrical settings, we consider two examples of fundamental nature. The radial cylindrical "Coulombic"Ṽ (ρ) = −2/ρ and the "harmonic oscillator"Ṽ (ρ) = a 2 ρ 2 /4. The spectral signatures of differentṼ (z) settings on the Coulombic and harmonic oscillator spectra are reported for impenetrable walls at z = 0 and z = L, for a Morse [31], for a non-Hermitian PT -symmetrized Scarf II [28,32,33], and for a non-Hermitian PT -symmetrized Samsonov [28,34] interaction models. Where, P denotes parity and T mimics the time reflection (cf., e.g., Ref. [28] and references cited therein on this issue). Our concluding remarks are in section 4.
to imply Using the above identities, one (with M (ρ, ϕ, z) ≡ M for simplicity of notations) may rewrite and Let use now consider a class of the mass functions defined as Which would, in effect, imply that the PDM Schrödinger equation [H − E] Ψ (ρ, ϕ, z) = 0 for Hamiltonian (1) be written as where At this point, one should notice that the choice of the mass function in (7) is inspired by the appearance of terms like (M ρ /M ), (M ϕ /M ), and (M z /M ) as multiplicities of the first-order derivatives in (8). This would, in fact, make the separability of (8) highly feasible and far less complicated. Moreover, following the traditional general wave function assumption, to ease coordinates separability of (8), we obtain It is obvious that separability is granted through a variety of choices. The simplest of which may be sought in an obviously "manifested-by-equation (12)" general identity of the form In this case, we may avoid any specifications on the forms of g (ρ), f (ϕ), and k (z) rather than being mathematically and quantum mechanically "very well" defined. However, the energy term, 2g (ρ) f (ϕ) k (z) E, in (12) suggests three feasible separabilities for f (ϕ) = 1 = k (z), k (z) = 1 = g (ρ), and f (ϕ) = 1 = g (ρ). We focus on one of these cases in the sequel.
Let us consider the position-dependent-mass function to be only an explicit Under these settings, equation (12) collapses into a simple separable form imply that and In due course, the solution of (15) reads Φ (ϕ) = exp (imϕ) where m = 0, ±1, ±2, · · · is the magnetic quantum number and Φ (ϕ) satisfies the single valued condition Φ (ϕ) = Φ (ϕ + 2π). Moreover, we obtain Consequently, one may cast and In the following section, we considerṼ (ρ) to represent a "Coulombic" and a "harmonic oscillator" and find the spectral signatures of differentṼ (z) potentials of (18) on the over all spectra.
3 Two examples; the radial cylindrical Coulombic and the harmonic-oscillator A priori, we remove the first-order derivative in the radial cylindrical part of (19) and redefine to obtain In fact, this 1D radial cylindrical Schrödinger equation provides an effective tool to study the effect of differentṼ (z) settings of (18) on the spectra of two interesting models of fundamental nature. The Coulombic and the harmonic oscillator [30]. Of course, such effects could be tested for other models.

Spectral signature of impenetrable walls at z = 0 and z = L
Lets us now consider that the above mentioned position-dependent-mass particle is trapped to move between two impenetrable walls at z = 0 and z = L. We may then takeṼ Consequently, equation (18) reads where Z (z) satisfies the boundary conditions Z (z = 0) = 0 = Z (z = L) and implies that Hence, K 2 z = n 2 z π 2 /L 2 and the quantum PDM particle here is quasi-free in the zdirection (i.e.,Ṽ (z) = 0) but constrained to move between the two impenetrable walls at z = 0 and z = L. The spectral signature of such z-dependent potential settings is clear, therefore. That is, a quantum particle endowed with a positiondependent-mass M (ρ, ϕ, z) = M (ρ) = ρ −2 and subjected to an interaction potential of the form withṼ (z) defined in (25), would admit exact energy eigenvalues given by On the other hand, a quantum particle endowed with a position-dependent- withṼ (z) defined in (25), would be accompanied by exact energy eigenvalues of the form
We may then closely follow the methodical proposal of Chen [31] to obtain where one should consider 2m = = 1, a → ǫ, E → K 2 z and x → z of Chen [31] to match our settings in (18). Therefore, a PDM quantum particle endowed with M (ρ, ϕ, z) = M (ρ) = ρ −2 and subjected to an interaction potential of the would admit exact energy eigenvalues given by Obviously, the condition √ D/ǫ −ñ z − 1 2 > 0 is manifested here and ought to be enforced, otherwise complex pairs of energy eigenvalues are obtained in the process. Moreover, a quantum particle with M (ρ, ϕ, z) = M (ρ) = ρ −2 subjected to an interaction potential would indulge the exact energy eigenvalues

PT -symmetrizedṼ (z) spectral signatures
We may now consider a PT -symmetrizedṼ (z) Scarf II in (18) so that where the corresponding Hamiltonian is known to be a non-Hermitian PTsymmetric Hamiltonian that admits exact eigenvalues (cf., e.g., Mustafa and Hence, a quantum particle with M (ρ, ϕ, z) = M (ρ) = ρ −2 moving in would encounter complex pairs of energy eigenvalues since K z = i n z + 1−A 2 (i.e., E nρ,m,nz ∈ C). Whereas, when the same PDM-particle is moving in it would admit exact and real energy eigenvalues as with K 2 z defined in (38). Of course, this should never be attributed to PTsymmetricity or non-PT -symmetricity of the original Hamiltonian (1) with the attendant complex non-Hermitian settings. It is very much related to the nature of separability we followed in this methodical proposal.
One may wish to consider the PT -symmetric Samsonov [34,28] model in (18). In this case Z (−π) = Z (π) = 0 and with a missing state n z = 2 (the reader may refer to Samsonov [34] on more details on this missing state). Hence, for a PDM quantum particle endowed with M (ρ, ϕ, z) = M (ρ) = ρ −2 and subjected to an interaction potential of the the exact energy eigenvalues would read Whereas, for the exact energy eigenvalues would read whereñ z = 1, 3, 4, · · · .

Concluding remarks
The kinetic energy operator in the PDM Hamiltonian (1) is a problem with many aspects that are yet to be explored. In the current work, we tried to study this problem within the context of cylindrical coordinates (ρ, ϕ, z). In due course, the essentials related with the kinetic energy operator in (1) (21) and admit exact energy eigenvalues documented in (23) and (24), respectively.
Nevertheless, the appearance of K z and K 2 z in (23) and (24) (46)). In fact, under such azimuthal symmetrization andṼ (z) setting, this set of exactly-solvable models may grow up as long as one can find exactly-solvable radially cylindrical models (hereby, exact-solvability may even include numerically exactly-solvable models as well). The recipe as how to collect the energy eigenvalues is clear in the above methodical proposal.