The simple harmonic oscillator is a topic of utmost importance on quantum mechanics. It is applicable in different physical situations and has the great advantage that has a closed form solutions for energy eigenvalues and eigen functions. Various investigations of the harmonic oscillator has been considered, the time-dependent harmonic oscillator, harmonic oscillator with point interaction, relativistic harmonic oscillator, an-harmonic oscillator, and the spiked harmonic oscillator.

    Recently, there has been some renewed interest in confined systems like the hydrogen atom and harmonic oscillator. This interest is highly motivated by the technological advances in the filed of semiconductor quantum dots, where the computations of electronic structure of such systems necessarily has to take into account the presence of finite boundaries on the system.

    The purpose here is to extend the isotropic harmonic oscillator plus a central potential N dimensions. Selected cases of the form of the central potential needs further work. The motivations behind this proposed work is due to its interest in atomic and molecular physics. One can also apply this work to psedo potentials in higher dimensions which is useful in nuclear physics. I addition, this is of great interest to mathematicians and physicists in theories studying so-called supper spaces or sipper-manifolds. This is related to Clifford analysis which allows for an elegant formulation of higher dimensional theories.