Sheared potentials and travelling nodes

TL;DR

This paper investigates how the energy spectrum of a one-dimensional Schrödinger problem responds to a deformation of split (shared) potentials under the constraint that the distance between classical turning points remains fixed. By analyzing two exactly solvable models—the split harmonic oscillator and the split linear potential—it demonstrates that eigenvalues oscillate as the deformation parameter changes, and these oscillations are tied to the migration of wavefunction nodes across the origin, i.e., a node crossing $x=0$ yields recurrences to the undeformed spectrum. The authors derive explicit conditions for the deformation values $\nu_{ij}$ at which $E_n(\nu_{ij})=E_n(1)$, showing that in the harmonic case $E_n(\nu_{ij})=\hbar\omega(n+\tfrac{1}{2})$, while in the linear case the energies closely track $E_n(1)$ but with model-dependent offsets, and asymptotically scale as $(2i+2j-1)^{2/3}$. The work provides a conjectural, node-dynamics-based mechanism explaining oscillatory spectral features in deformed, piecewise potentials, with potential relevance to related analyses of isoperiodic or similar potentials.

Abstract

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.