An elementary, analytically soluble example of a pair of isospectral potentials is discussed. This example was developed in response to an inquiry by an undergraduate student who was intrigued by the widely used (but not often discussed in undergraduate courses) Rydberg–Klein–Rees (RKR) procedure for inverting rovibrational data to determine potential functions of diatomic molecules, which is based on the semiclassical (WKB) approximation. The pair of potentials discussed provide a clear elementary example of the insufficiency of the energy level spectrum to determine the potential, even in a case in which the bound states form a complete set.
I. INTRODUCTION
For nearly fifty years, I started each of my courses preaching about the benefits of selfstudy ahead of class, referring to the introduction of book printing by Gutenberg as the point in history when the role of the professor should have transformed from a knowledge provider to guide in an adventurous intellectual pursuit. For the majority of my students, these sermons remained futile.
The recent abrupt need to switch to remote teaching/learning allowed a radical transformation of my teacher–student mode of interaction, remotely teaching an elective undergraduate Molecular Spectroscopy course offered to second and thirdyear Materials Science and Engineering students who had completed a onesemester quantum chemistry course at the GuangdongTechnion Israel Institute of Technology in Shantou, Guangdong, China. Rather than mimicking traditional lecturing via live (“zoom”) or videorecorded classes, the students were offered a (closed) website (“moodle”) where they could find reading materials, references to appropriate chapters in their course textbook,^{1} and exercises, along with a detailed learning guide accompanying each chapter. The students were promised credit for “good” questions and comments, and I made an effort to respond promptly (synchronizing my day with China time—six hours ahead of Israel), loading crudely edited questions and responses on the course website. Indeed, a steady flow of enquiries required my attention, providing me with feedback that I have never before benefitted from about the students' progress and their difficulties.
II. INVERTING SPECTRAL DATA: INSUFFICIENCY OF THE VIBRATIONAL SPECTRUM
The Rydberg–Klein–Rees (RKR) procedure for inverting vibrationrotation data to obtain the potential energy function of a diatomic molecule is briefly discussed in the textbook used in the course mentioned above^{1} (p. 216). The nonuniqueness of the inversion of vibrationalonly data into a potential function intrigued the curiosity of one of the students (see acknowledgements). I quickly offered an appropriate source,^{2} supplemented by an elementary introduction to the WKB (semiclassical) approximation, which the RKR procedure is based on. The purpose of the present comment is to illustrate that, in a particularly simple context, vibrational energy levels are not sufficient to determine the potential. In more fancy language, to illustrate the existence of isospectral potentials (e.g., Ref. 3).
We will only consider potentials, $V(x),\u2009\u2212\u221e<x<\u221e$, with a single minimum at some x_{e}, i.e., such that V(x) is monotonically decreasing over $\u2212\u221e<x< x e$ and monotonically increasing over $ x e<x<\u221e$. For energies that satisfy $E\u2265V( x e)$, we will refer to the two roots of the equation V(x) = E as the left and right classical turning points, $ x \u2113$ and x_{r}. If $ lim x \u2192 \xb1 \u221e V ( x ) = V \xb1 \u221e < \u221e$, then we define $ x \u2113 = \u2212 \u221e$ for $ E \u2265 V \u2212 \u221e$ and $ x r=\u221e$ for $E\u2265 V \u221e$.
The following definitions and facts are provided:

The classical periods of oscillation in a onedimensional harmonic (parabolic) potential do not depend on the energy (or amplitude) of the oscillation. Potentials that satisfy this property are referred to as isochronous.

Two potentials, $ V ( 1 )(x)$ and $ V ( 2 )(x)$, which have the same periods of oscillation at all energies, are referred to as isoperiodic.

Two potentials that have the same distance between the right and the left classical turning points $ x \u2113(E)\u2212 x r(E)$, at all energies, are referred to as shear equivalent. Shear equivalent potentials are isoperiodic.^{4,5}

Potentials that are shearequivalent to a parabola are isochronous.^{6,7} A much broader view of isochronous systems was offered by Calogero.^{8}

A potential, which is continuous with continuous first and second derivatives (a C^{2} potential), is isochronous if and only if it is shearequivalent to a parabola.^{9,10} However, singular isochronous potentials have been reported.^{11}

Shearequivalent potentials have equal Bohr–Sommerfeld phase integrals, $ \u222e pdx$, at all energies. It follows that such potentials have the same EBK (or lowest order WKB) spectra, except in limiting cases in which the Maslow index has to be modified, as illustrated below.

Shear equivalent potentials need not share a common quantum spectrum.^{12}
In the RKR procedure (which is based on the semiclassical quantization), the vibrational spectrum provides the distances between the left and right “classical turning points” ( $ R \u2113$ and R_{r}, respectively) at the energies of the available vibrations. This is not enough to construct the potential. The task of determining the potential is completed by examining the rotational levels associated with each vibrational level. They provide a distinct moment of inertia at each vibrational level, from which a “bondlength” is derived, which is (naively) assumed to be equal to $ R \u2113 + R r/ 2$.
III. A PAIR OF ISOSPECTRAL POTENTIALS
What about quantum mechanics?
That was easy! But what about the second potential?
Well, semiclassically, it ought to be isospectral. For the harmonic oscillator, the semiclassical treatment yields the same spectrum of equidistant levels as the solution of the Schrodinger equation (except that in the semiclassical treatment, the energy is given in the form of $\u210f\omega n$—the form originally postulated by Planck—rather than $\u210f\omega n + 1 2$, unless we invoke the Maslow correction). So one might guess that the same would hold true for $ V ( 2 )(x)$.
So $ V ( 2 )(x)$, just like $ V ( 1 )(x)$, has a spectrum of equidistant levels, separated by $\u210f\omega $. To make these two potentials fully isospectral, we need to pull $ V ( 2 )(x)$ down by $ \u210f \omega / 4$, but this does not affect the wavefunctions or anything that depends on them.
We note (without attempting an interpretation) that these electric dipole matrix elements rise with n somewhat more slowly than those for the symmetric harmonic potential, $ V ( 1 )$.
IV. CONCLUSION
The comparison between the harmonic oscillator and the halfoscillator spectra provides an elementary illustration of isospectrality. Examination of the diagonal and the offdiagonal dipole matrix elements suggests that the vibrationrotation spectra as well as the vibrational transition probabilities can be used to distinguish between such isospectral potentials.
The author's “hidden agenda” is to illustrate that the transition to remote teaching/learning imposed by the Covid19 pandemic can possibly revolutionize the professor's role from an information provider to a guide in an intellectual pursuit. This requires an intensively involved teaching style that university administrators who look for cheaper massteaching technologies will not be happy with, as well as curious and committed students.
ACKNOWLEDGMENTS
The author wish to thank Mr. Zhixiang YAO, second year MSE student at GTIIT, for his curiosity, which inspired the present paper. Professor Ed Montgomery kindly read the paper and provided helpful advice.
APPENDIX: Generating functions for harmonic oscillator matrix elements
This appendix is presented as an opportunity to introduce a somewhat richer application of generating function ideas than are usually offered to undergraduates.
The various matrix elements reported above were calculated by explicit integration (over $0\u2264x<\u221e$) of appropriate expressions in terms of harmonic oscillator wave functions.
The oldfashioned application of such generating functions involved somewhat tedious expansion of the generating function $ \pi / 2 exp (2st)(1+erf(s+t))$, a task that a symbolic programming platform such as maple readily performs. We note that the expansion contains no terms with $ n 1\u2260 n 2$ when they are both odd (since they correspond to distinct eigenfunctions of the “halfoscillator” Hamiltonian), but it does contain spurious terms with either n_{1} or n_{2} or both being even.