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 5 V ( x ) |$_{x}$$_{→}$$_{z}$ = [ V ( − x )] ∗ |$_{x}$$_{→}$$_{z}$ = V $^{∗}$( − z $^{∗}$) . (7) V ( − iy ) = V $^{∗}$( − iy ) ∈ R = ⇒ V ( y ) = V $^{∗}$( y ) ∈ R , (8) m = m$_{±}$ = ⇒ m =     m$_{+}$ = + | m | > 0 m$_{−}$ = − | m | < 0 (9) H$_{x}$ = H$_{x}$$_{;}$$_{m}$$_{+}$ = p 2 x 2 m$_{+}$ + 1 2 m$_{+}$ω $^{2}$x2 (10) H$_{y}$$_{;}$$_{m}$$_{−}$ = p 2 y 2 m$_{−}$ + 1 2 m$_{−}$ω $^{2}$y2 ∈ L $^{2}$( R ) = H . (11)