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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 form 

the exact energy eigenvalues would read 

Whereas, for 

the exact energy eigenvalues would read 

where ˜ n$_{z}$ = 1 , 3 , 4 , · · · . 

## 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) are reported. The separability of the Schr¨ odinger equation is sought through a radial cylindrical position dependent mass M ( ρ, ϕ, z ) = M ( ρ ) = 1 /ρ 2 accompanied by an az- imuthally symmetrized interaction potential V ( ρ, ϕ, z ) = ρ 2 [ ˜ V ( ρ ) + ˜ V ( z ) ] / 2, 

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