Correspondence between the q-deformed harmonic oscillator and finite range potentials

The q-deformed harmonic oscillator is revisited in connection with the spectrum of finite range potentials. It is found that the finite series expansion of the canonical variables, the q-deformed phase space coordinate and momentum variables, generates a local momentum dependent interaction. It turns out that the resulting spectrum exhibits features of the spectrum of a finite range potential, added to the low-lying harmonic oscillator behavior. Thus, the otherwise unbounded spectrum of the harmonic oscillator behaves, in the q-deformed version, as the spectrum of a finite range potential subject to specific boundary conditions