Time dependent rationally extended Poschl-Teller potential and some of its properties

D. Nath; P. Roy

REVIEW 2 major objections 5 minor 47 references

Time dependent rationally extended Poschl-Teller potential and some of its properties

T0 review · 2 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The paper constructs exact solutions of the time-dependent Schrödinger equation with oscillating boundaries for the Pöschl-Teller potential and its rationally extended supersymmetric partner, in terms of X1 Jacobi exceptional orthogonal…

desk verdict A useful but incremental exact-solution construction in Sec. 2; the average-energy section, however, contains an algebraic omission that produces spurious imaginary energies and needs to be reworked. read the letter →

arxiv 2009.12851 v2 pith:JTDI4EPI submitted 2020-09-27 math-ph math.MPquant-ph

D. Nath , P. Roy This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 81Q0581Q6033C4535Q41

keywords exceptionalorthogonalpolynomialstime-dependentSchrödingerequationoscillatingboundaryPöschl-TellerpotentialsupersymmetricquantummechanicsseparationofvariablesJacobimovingproblem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the time-dependent Schrödinger equation with an oscillating Dirichlet boundary can be solved exactly when the fixed-boundary problem is the Pöschl-Teller well or its rationally extended supersymmetric partner. A separation-of-variables ansatz with a time-dependent coordinate map reduces the moving-boundary equation to a fixed eigenvalue problem, and the resulting wavefunctions are given in closed form, with the partner sector expressed through X1 Jacobi exceptional orthogonal polynomials. The authors then compare the two sectors across two oscillating boundary profiles, computing probability densities, average energies, position and momentum spreads, and Heisenberg uncertainty products. The paper's claim is that these exact solutions hold for all times and that the rationally extended sector shows smaller root-mean-square widths and lower uncertainty products.

What carries the argument

The load-bearing device is the separation ansatz $\psi(q,t)=e^{\Phi(q,t)}Q(q)T(t)$ together with the moving-coordinate map $q=\pi(x-L(t)/2)/L(t)$. The phase $\Phi$ is fixed by $a(t)=\frac{i}{2}L_1(t)\dot{L}_1(t)$, $b(t)=\frac{i}{2}L_1(t)\dot{\alpha}(t)$ and a carefully chosen $c(t)$, so that the time-dependent equation separates into $-\frac{d^2Q}{dq^2}+\tilde{V}(q)Q=\epsilon Q$ and $T(t)=e^{-i\epsilon\tau(t)}$ with $\tau(t)=\int_0^t ds/L_1(s)^2$. The potentials come from the superpotential $\tilde{W}(q,A,B)=(-B-\frac12)\tan q+(A-\frac12)\sec q+\frac{2B\cos q}{2A-1-2B\sin q}$, which produces the Pöschl-Teller partner $\tilde{V}^{(-)}$ and the rationally extended partner $\tilde{V}^{(+)}$; the latter's eigenfunctions involve the operator $\hat{O}^{(\alpha,\beta)}$ acting on Jacobi polynomials, defining the X1 Jacobi exceptional orthogonal polynomials. This reduction converts an oscillating-boundary problem into fixed-boundary spectral data.

What would settle it

Substitute the explicit $\psi_n^{(+)}(x,t)$ from Eq. (28) into the time-dependent Schrödinger equation (1) with $V^{(+)}(x,t)$ from Eq. (27) and the boundary condition, or equivalently evaluate the overlap integral $\int_0^{L(t)} \Omega^{(+)}[\Phi_n^{(+)}]^2 dx$ for a chosen parameter set such as $A=5, B=3.4$ and check whether the constant in Eq. (32) equals one; if it does not, recompute the moments with that constant divided out and see whether the plotted average energies and uncertainty comparisons change.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the moving-boundary equation (1), together with the boundary conditions $\psi(0,t)=0=\psi(L(t),t)$, admits the exact factored solutions $\psi_n^{(-)}(x,t)$ and $\psi_n^{(+)}(x,t)$ of Eqs. (22) and (28). The construction changes variables to $q=\pi(x-L(t)/2)/L(t)$, writes the wavefunction as $e^{\Phi(q,t)}Q(q)T(t)$, and chooses $\Phi$ and the potential's auxiliary terms so that space and time separate. The space eigenfunctions $Q_n^{(-)}$ and $Q_n^{(+)}$ belong to the Pöschl-Teller potential and to its supersymmetric partner, whose solutions are built from the exceptional polynomials $\hat{P}^{(\alpha,\beta)}_{n+1}(z)$. The paper reports that the probability densities localize periodically with the boundary motion, that the real parts of the average energies of the two sectors are nearly identical while the imaginary parts differ, and that the $(+)$ sector has smaller RMS spreads and lower uncertainty products for the states and parameters examined.

Load-bearing premise

The average-energy and moment calculations in the $(+)$ sector assume the wavefunctions are normalized to unity, but the overlap integral in Eq. (32) carries a normalization factor that is not one, so if that factor is not divided out, the quoted normalized observables are off by a constant.

Editorial extensions

If this is right

For every choice of $L(t)$ with the assumed regularity, the formulas produce a new exactly solvable moving-boundary system, so the two profiles in the paper are examples of a much larger family rather than isolated cases.
The explicit solutions can serve as testbeds for numerical methods for time-dependent Schrödinger equations with moving boundaries, since the exact answer is known at arbitrary times.
The paper's comparison indicates that switching from the standard Pöschl-Teller well to its rationally extended partner changes the shape of the density but keeps the real average energy nearly unchanged, while lowering the uncertainty product; this gives a concrete example of how exceptional-polynomial potentials alter localization properties.
The factorization into time-independent moments $I_1, I_2, I_3$ times $L(t)/\pi$ means the uncertainty inequality (43) holds for the whole family once the fixed-boundary integrals satisfy it, so the moving-boundary setting does not introduce extra sources of violation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the normalization factor implied by Eq. (32) is divided out before the moments $I^{(+)}_{k,n}$ are used, the average energy values for the $(+)$ sector will be rescaled; the paper's qualitative conclusions about near-identical real parts and different imaginary parts might survive, but the quantitative comparison should be checked.
The same separation-plus-supersymmetry pipeline should apply to other rationally extended potentials solved by $X_l$ or multi-index exceptional orthogonal polynomials, yielding further exact moving-boundary families by replacing the partner eigenfunctions.
Because the potentials here reduce to fixed-boundary data, the construction suggests a general recipe: any exactly solvable fixed-boundary potential can be lifted to an oscillating-boundary potential with the same phase ansatz, making the present paper a template rather than a single example.
A natural test would be to let both boundaries move independently; the current ansatz assumes a symmetric moving interval, and it is not obvious that the factorization survives asymmetric motion.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The separation-of-variables construction in Sec. 2 is the honest core of this paper, and it largely works. The authors build time-dependent versions of the Pöschl–Teller potential and its rationally extended partner, with oscillating boundaries, and write down exact wavefunctions in terms of X1 Jacobi exceptional polynomials. The potentials in Eqs. (21) and (27) and the wavefunctions in Eqs. (22) and (28) are plausible solutions of the stated moving-boundary problem. That is a genuine, if incremental, extension of earlier work; the paper itself cites Ref. [39] for time-dependent exceptional potentials, so it is not claiming a wholly new mechanism.

The soft spot is Sec. 3, and the stress-test note lands. Eq. (46) does not follow from Eq. (44): the moving-coordinate correction involving A ∂F/∂q is dropped. Once that term is included, the imaginary part cancels, consistent with the fact that a normalized solution of a real-potential TDSE with Dirichlet boundaries should have real average energy. So the complex average energies in Fig. 3 are artifacts of the omission, not physical. There is also a normalization issue that is independent and compounding. Eq. (26) gives ∫ΩΦ_nΦ_m dx = δ[n(N)^2], not 1, and Eq. (32) is similarly not normalized to unity. That means every expectation value and RMS quantity in Sec. 3 needs to be divided by the actual normalization factor. This is not a cosmetic typo; it changes the reported numbers.

What is done well: the potential construction is deterministic, the references to the static POT and exceptional polynomial literature are appropriate, and the probability-density and uncertainty plots are the kind of qualitative output one would expect from these solutions. The citation pattern looks fine; no signs of over-referencing or circular support.

Who should read this: someone working on moving-boundary quantum problems or time-dependent extensions of exactly solvable potentials will find useful building blocks. The reader will need to redo Sec. 3 before trusting the quantitative claims. The construction itself deserves referee time, not a desk reject, but the paper should be returned for major revision rather than accepted in its current form. I would not cite the average-energy results as they stand.

Referee Report

2 major / 5 minor

Summary. The paper constructs time-dependent versions of the Pöschl-Teller potential and its rationally extended supersymmetric partner, using the separation-of-variables method for a moving boundary L(t). The wavefunctions are expressed in terms of X1 Jacobi exceptional orthogonal polynomials, and the paper derives exact solutions for the oscillating-boundary problem. It then computes probability densities, average energies, RMS widths in position and momentum, and Heisenberg uncertainty products for both potentials, for two explicit profiles of L(t). The central positive claim is that Eqs. (22) and (28) exactly solve the time-dependent Schrödinger equation (1) with the potentials (21) and (27) and Dirichlet boundary conditions on [0,L(t)].

Significance. If the Sec. 2 construction is correct, the paper supplies a rare exact solution family for a moving-boundary problem whose static counterpart is solvable by exceptional orthogonal polynomials. The construction is deterministic and parameter-free in the sense that no data fitting is involved, and the paper is explicit about the gauge choice and the normalization identities (26) and (32), which makes the derivation checkable. The main value lies in the explicit potentials and wavefunctions, and in the comparison of localization and uncertainty properties between the ordinary and the exceptional sectors. However, the quantitative predictions in Sec. 3 are compromised by an algebraic omission in the average-energy derivation and by missing normalization factors, so the numerical figures that summarize the physical properties do not currently support the claims made from them.

major comments (2)

[Sec. 3, Eqs. (44)-(47)] Equation (46) does not follow from Eq. (44). Writing A = (dot α + q dot L1)/L1, the right side of Eq. (44) contains the term + ∫ Q^2 A F_q dq after the integration by parts of the A Q Q' term. Using F_q = (1/2) L1 dot L1 q + (1/2) L1 dot α (with the corrected first term in Eq. (45), which should be (1/4) L1 dot L1 q^2, not (1/(4L1)) dot L1 q^2), this correction contributes + (1/2) dot α^2 + dot α dot L1 q + (1/2) dot L1^2 q^2 to the real part and makes the two imaginary terms - (i/2) ∫ ∂_q A Q^2 dq and - i (dot L1/(2 L1)) ∫ Q^2 dq cancel exactly. The coefficients h0, h1, h2 displayed in Eq. (47) match only - ∂_t F, i.e. they omit the A F_q term. Hence Eq. (46) is not the value of the integral in Eq. (44), and the conclusion that the average energy is 'complex in general' in the sentence after Eq. (47) is an artifact of this omission. For a real potential and normalized states with Dirichlet boundary conditions, i ∫ ψ* ∂_t ψ dx equals ⟨ψ|H|ψ⟩ and must be real, so the imaginary parts shown in Fig. 3 are unphysical and should be replaced by zero once the calculation is corrected.
[Sec. 3, Eqs. (26), (32), (41)-(47)] The expectation values and RMS formulas assume that the q-space wavefunction Q(q) is normalized to unity, but the paper's own normalization identities show that the states as defined are not normalized. Equation (26) gives ∫ Ω_- Φ_n Φ_m dx = [N_n]^2 δ_nm, and Eq. (32) gives ∫ Ω_+ Φ_n Φ_m dx = (E_n/4) [(β-α)(β+n) N_n]^2 δ_nm, neither of which is 1 for n=m. The moments I_k,n defined in Eq. (47) are raw integrals of [Q_n]^2 q^k, with no division by the norm squared. Unless Q_n in Sec. 3 is explicitly redefined to include the normalization factor, all I_k,n and hence all RMS values, momentum RMS values, uncertainty products in Eqs. (41)-(43), and the normalized average energy in Eq. (46) are incorrect for both sectors, and especially for the (+) sector whose norm is not even N_n^2. This affects every quantitative claim in Figs. 3-6 and must be repaired before the reported properties can be accepted.

minor comments (5)

[Eq. (45)] The first term should be (1/4) L1 dot L1 q^2 rather than (1/(4 L1)) dot L1 q^2; with the printed form the expression is inconsistent with the phase factor in Eq. (25) and with the units of the other terms.
[Eq. (44)] The notation i dot F(t) is misleading because F is a function of q and t; the term should be i ∂_t F(q,t), and the subsequent derivation of Eq. (46) should keep the q-dependence of F_q explicitly.
[Eq. (43), Fig. 1, Fig. 3 captions] There are several typographical slips: in Eq. (43) the superscript on I_1,n is broken, in the Fig. 1 caption 'A, = 5,B = 0.2' should be 'A = 5, B = 0.2', and in Fig. 3 the labels for the imaginary parts are inconsistent with the text.
[Eqs. (26) and (32)] The normalization constants N_n in Eqs. (26) and (32) are not connected explicitly to the normalization factor N_n^{(A,B+1)} defined in Eq. (20); stating the relation between Φ_n, Q_n, and the normalized wavefunction would remove the ambiguity discussed in the major comments.
[References] Reference [29] has an incomplete page range and inconsistent formatting, and reference [31] is cited both as the separation-of-variables method and as the source of the gauge choice; a consistent citation format would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the moving-boundary construction is self-contained and the cited static results are independent inputs.

full rationale

The paper derives the time-dependent Pöschl–Teller and rationally extended Pöschl–Teller potentials through an explicit separation-of-variables ansatz, ψ = e^Φ Q(q)T(t), with Φ fixed by Eqs. (7)–(8) and Q, E obtained from the static Schrödinger equation (11). The static superpotential and X1 Jacobi exceptional polynomial solutions are taken from Quesne [40], an external independent source, and the separation technique from Efthimiou–Spector [34]; neither source incorporates the moving-boundary result claimed here. No parameter is fitted to any data subset, and no output is made equal to an input by definition: the time-dependent potentials (21), (27) and wavefunctions (22), (28) are generated by substituting the static solutions into the moving-boundary framework, and the boundary conditions are enforced by the coordinate transformation (3). The only author self-citations, e.g. [25] and [38], are background references and are not load-bearing for the central derivation. The expectation-value section contains apparent algebraic and normalization issues—Eq. (32) is not a unit-normalization condition, and Eq. (46) as printed does not reproduce Eq. (44) when the A F_q term is retained—but these are correctness defects, not circular reasoning, because they do not reduce any claimed prediction to an input by construction. Consequently, no circular step is exhibited and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

No new entities are introduced. The free parameters A, B, A1, B1, ω are model parameters chosen for plots, not fitted to data. The axioms are standard background results or the gauge choices of the separation method. The main concern is the unverified normalization of the (+) sector states.

free parameters (3)

A, B = A=5, B=0.2 and B=3.4 in different figures
Parameters of the superpotential inherited from the static rationally extended Pöschl-Teller potential. Chosen by hand to plot potentials and observables; the exact-solution construction holds for any allowed values.
A1, B1, ω (in L(t) profile 2) = A1=1, B1=0.5, ω=1 in figures
Parameters in the oscillating boundary L(t)=A1π√(1+B1 cos ωt). Chosen by hand for numerical illustration; they do not affect the existence of exact solutions.
g0(t)
An arbitrary function appearing in the potential (Eq. (13)). Not specified further; effectively a free gauge choice that does not affect the solution structure.

assumptions (3)

domain assumption The static rationally extended Pöschl-Teller potential and its partner are exactly solvable with the given superpotential and Jacobi exceptional polynomials (from Ref. [40]).
The paper relies on the prior derivation of the static potential and its eigenfunctions, equations (14)-(19), without re-deriving them.
ad hoc to paper The separation-of-variables technique for moving boundaries, including the gauge choice g(t)L1^2(t)=1, applies to this potential.
The paper assumes the ansatz (4)-(5) and the specific choices for U(q,t) and the phase factor lead to a separated time-independent problem. This is the standard method of Ref. [34] applied here.
domain assumption The wavefunctions are properly normalizable under the given parameter restrictions (A > Max{B+1.5, |B|+0.5}).
The paper imposes this condition to avoid singularities, but the normalization of the (+) sector states is not verified to be unity (see weakest_assumption).

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Pith. "Pith review of Time dependent rationally extended Poschl-Teller potential and some of its properties." pith.science (2026). https://pith.science/paper/JTDI4EPI

@misc{pith2026200912851,
  author       = {Pith},
  title        = {Pith review of: Time dependent rationally extended Poschl-Teller potential and some of its properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JTDI4EPI}},
  note         = {Machine review of arXiv:2009.12851}
}

read the original abstract

We examine time dependent Schrodinger equation with oscillating boundary condition. More specifically, we use separation of variable technique to construct time dependent rationally extended Poschl-Teller potential (whose solutions are given by in terms of X1 Jacobi exceptional orthogonal polynomials) and its supersymmetric partner, namely the Poschl-Teller potential. We have obtained exact solutions of the Schrodinger equation with the above mentioned potentials subjected to some boundary conditions of the oscillating type. A number of physical quantities like the average energy, probability density, expectation values etc. have also been computed for both the systems and compared with each other.

Figures

Figures reproduced from arXiv: 2009.12851 by the authors.

**Figure 1.** Plots of the potential V (−) (x, t) (solid curve) with V (+)(x, t) (dashed curve), for (A) L(t) = π(2 + sin t) and (B) L(t) = √ A1π 1+B1 cos ωt and A, = 5, B = 0.2, a = q2 3 , b = √ 2, ω = 1. To avoid the singularity we consider A > M ax {B + 1.5, |B| + 0.5}. Next, we examine the instantaneous probability densities ρ ± n = |ψ ± n (x, t)| 2 at different times. A plot of the instantaneous densities are given in [PITH… view at source ↗

**Figure 2.** Plots of probability densities ρ (−) n (x, t) (solid curve) with ρ (+) n (x, t) (dashed curve), for (A) n=0, (B) n=0, (C) n=1, (D) n=1, (E) n=2, (F) n=2. The left column is defined for L(t) = π(2 + sin t) whereas the right column is defined for L(t) = √ A1π 1+B1 cos ωt . The other parameters are taken as A, = 5, B = 0.2, a = q2 3 , b = √ 2, ω = 1. We now examine the behavior of the average energy E¯. It can be seen … view at source ↗

**Figure 3.** Comparison of the average energy: (1) (A)-(B): rea [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Comparison of the RMS in position space (A)-(B) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Variation of the uncertainty in position space (so [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Comparison of the uncertainty product for (A) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [39]

Schulze-Halberg and B

A. Schulze-Halberg and B. Roy, J. Math. Phys. 55 (2014) 123506

work page 2014

[2] [1]

J. R. Ray, Phys. Rev. A 26 (1982) 729-733

work page 1982

[3] [2]

M. J. Engleﬁeld, J. Phys. A: Math. Gen. 20 (1987) 593-600

work page 1987

[4] [3]

I. H. Duru, J. Phys. A: Math. Gen. 22 (1989) 4827-4833. 11

work page 1989

[5] [4]

V. V. Donodov, V. I. Manko, D. E. nikonov, Phys. Lett. A 162 (1992) 359-364

work page 1992

[6] [5]

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work page 1996

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F. Finkel, A. Gonzalez-L´ opez, N. Kamran, M. A. Rodrguez, J. M ath. Phys. 40 (1999) 3268-3274

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