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for simplicity of notations) 

This would in turn imply that 

where V ( y ) is a real-valued function, therefore. Some illustrative examples can be found section 6 of [46]. 

In this respect, a remedy for the t’ Hooft and Nobbenhuis [44] harmonic oscillator Hamiltonian above may be sought in a mass parametrization recipe accompanied with the de-Sitter and anti-de-Sitter transformation x ∈ R −→ iy ∈ i R . That is, 

Such a mass parametrization would, in turn, suggest that the t’ Hooft and Nobbenhuis [44] harmonic oscillator 

in (2) with m$_{+}$ = − m$_{−}$ reads 

In this case both H$_{x}$$_{;}$$_{m}$$_{+}$ and H$_{y}$$_{;}$$_{m}$$_{−}$ are isospectral and both admit normalizable eigenfunctions. For example, the ground state in x -space ∼ exp ( − m$_{+}$ωx $^{2}$/ 2 ) and that in the y -space ∼ exp ( − m$_{−}$ωy $^{2}$/ 2 ) are both normalizable. The mass 

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