Exactly solvable new classes of potentials with finite discrete energies

In this work, we propose more realistic models with discrete and finite number of energy levels that could fit well to molecules with potentials that were modeled previously as harmonic oscillator. The considered potentials could be also used as good models in quantum physics, statistical and condensed matter physics, atomic physics, nuclear physics, particle physics, high energy physics, mathematical physics, as well as in chemistry of complex molecules. More precisely, we derive the solutions of two families of Schrödinger equations using supersymmetric quantum mechanics technique for superpotentials having shape invariance properties, and where their eigenvalues and eigenfunctions are exactly determined. The range of their finite number of bound states is given explicitly. Furthermore, this result will contribute in extending the already small list of exactly solvable Schrödinger equations, where we have summarized in a table all well-known potentials having exact solutions and their superpotentials, their partner potentials, and their energies, as well as, the newly discovered potentials proposed here.