Sheared potentials and travelling nodes

When a sheared potential is deformed in such a way that the distance between the classical turning points remains constant the eigenvalues of the Schrödinger equation oscillate with respect to the potential parameter responsible for the deformation. We show that such an oscillation is intimately related to the passing of the nodes of the corresponding eigenfunctions through the origin. We illustrate this effect by means of the split harmonic oscillator and the split linear potential.