Exactly solvable Schrödinger eigenvalue problems for new anharmonic potentials with variable bumps and depths

A new approach based on Darboux transformations is introduced to generate classes of solvable Schrödinger equations for new anharmonic potentials with variable bumps and depths. By introducing the concept of a transformation key, we present a method of controlling the number of bumps and their depths in these potentials. Although this method was applied to the one-dimensional generalized harmonic oscillator potential, it can be easily adapted to generate exactly solvable potentials using other known quantum potentials.