The Dance of the Sheared Eigenfunctions

TL;DR

The paper investigates how quantum spectra change under a continuous shearing of one-dimensional potentials while preserving a classical isoperiodicity. It develops exact, piecewise solutions for two monomial families—one with a linear potential and one with a harmonic-oscillator potential—solving the Schrödinger equation with Airy and D-parabolic cylinder functions, and derives transcendental conditions that determine the spectra as functions of the shear parameter $\nu$. The authors show that, although classical isoperiodicity persists, the quantum spectra are not isospectral across the shear family, and they connect spectral shifts to the continuous deformation of the eigenfunctions via the Hellmann–Feynman theorem, including a force-based interpretation near $\nu\approx 1/2$. The work emphasizes that analyzing the eigenfunctions is essential to understanding spectral changes and outlines physical interpretations and potential applications to Stark-like effects and adiabaticity control in quantum systems, with future directions to quantify the energetic cost of shear and extend the approach to other power-law potentials.

Abstract

In this work, we delve into the theory of sheared potentials in non-relativistic quantum mechanics. After defining what we mean by a family of sheared potentials, we consider these families in two particular but emblematic cases, the harmonic oscillator and the symmetric potential well proportional to $|x|$. In both cases, besides determining the spectra, we analyse the impact of the shearing process on the respective eigenfunctions. The latter analysis is typically left aside in the literature, but here we show that the sheared eigenfunctions yield insights that allow for a deeper understanding of the main features exhibited by the spectra. Finally, we make a few comments about the connection between the change in the spectra of a given sheared family and the necessary work that must be made by an external agent to implement such a change.

Figures (8)

• Figure 1: Two potential wells belonging to the same family of sheared potentials. Note that the shearing condition, $b_{\xi'} - a_{\xi'} = b_{\xi} - a_{\xi}$, is fullfilled.
• Figure 2: Three sheared potentials of the family of the first monomial well. The shearing condition is also indicated in the figure with the horizontal double arrows.
• Figure 3: In the left panel we plot the normalized energy levels, $E_n(\nu)/E_n(1)$, for $n=0,1,2,3,4$, in terms of $\nu$. In the right panel we give a zoom in the plot of $E_2(\nu)$ in terms of $\nu$ to emphasize the oscillations for $\nu$ close to $1$.
• Figure 4: Complete eigenfunctions $\psi_\nu(x)$ for some values of $\nu$ for the first two energy levels (the fundamental state on the left and the first excited state on the right). In these plots, we considered $k=1$.
• Figure 5: In the left panel we plot the normalized energy levels for the sheared family of the harmonic oscillator, $E_n(\nu)/E_n(1)$, with $n=0,1,2,3,4$, in terms of $\nu$. In the right panel we give a zoom in the plot of $E_2(\nu)$ to emphasize the oscillations for $\nu$ close to $1$.