Multi-parameter isospectral Fokker-Planck equations

Choon-Lin Ho

REVIEW 2 major objections 7 minor 29 references

Multi-parameter isospectral Fokker-Planck equations

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Starting from any Fokker-Planck equation with constant diffusion and time-independent drift, this paper constructs an n-parameter family of partner equations with the same eigenvalues but different eigenstates, by deleting the lowest n…

desk verdict A standard SUSY construction extended to n parameters with explicit eigenstates; the formulas are likely right but two printed errors—wrong seed condition and wrong prefactor in Eq. (27)—need fixing before it is citable. read the letter →

arxiv 2506.12939 v2 pith:XHM6FJHB submitted 2025-06-15 math-ph cond-mat.stat-mechgr-qcmath.MPquant-ph

Choon-Lin Ho This is my paper

classification math-phcond-mat.stat-mechgr-qcmath.MPquant-ph MSC 35Q8481Q6035R11

keywords Fokker-PlanckequationisospectraltransformationDarboux-CrumMittag-LefflerfunctionfractionalWronskiandriftpotential

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 shows how to turn any Fokker-Planck equation with constant diffusion and time-independent drift into a whole family of partner Fokker-Planck equations that share the same energy spectrum but have different drifts and different eigenstates. The construction first removes the n lowest eigenstates by repeated Darboux-Crum transformations, then puts the same energy levels back with a reverse Darboux transformation, each level carrying a free parameter. The paper writes down all eigenstates of the deformed system explicitly and gives a formula that turns any solution of the original equation into a solution of the deformed one. It also extends the recipe to fractional Fokker-Planck equations, where exponential time decay is replaced by Mittag-Leffler relaxation, and it comments on an application to black-hole thermal potentials. A reader should care because the result supplies a method for generating many exactly solvable Fokker-Planck equations from any known one.

What carries the argument

The load-bearing object is the Darboux-Crum ladder: given eigenstates $\phi_0,\phi_1,\ldots$ of $H$, the n-th rung states are the Wronskian quotients $\phi_k^{(n)}=W[\phi_0,\ldots,\phi_{n-1},\phi_k]/W[\phi_0,\ldots,\phi_{n-1}]$, with the n-th drift $D^{(n)}=2(\ln|\phi_n^{(n)}|)'$. The reverse step uses virtual seed functions $\Phi_s$ defined recursively by $\Phi_s(\lambda_s,\ldots,\lambda_{n-1}) = B^\dagger_{s+1}\cdots B^\dagger_{n-1} A^{(n-1)}\cdots A^{(s+1)}\Phi_s(\lambda_s)$, where $\Phi_s(\lambda_s)=(I_s+\lambda_s)/\phi_s^{(s)}$ and $I_s=\int^x [\phi_s^{(s)}]^2$. The final ground state of the isospectral pair is $\Phi_0^{-1}(\lambda_0,\ldots,\lambda_{n-1})$, and the operator $L_n=B^\dagger_0\cdots B^\dagger_{n-1}A^{(n-1)}\cdots A^{(0)}$ maps original eigenstates $\phi_k$ with $k\ge n$ to isospectral eigenstates while annihilating the n lowest states; this is what makes Eq. (27) a direct recipe for converting solutions.

What would settle it

Compute $(A^{(0)})^\dagger \Phi_0(\lambda_0)$ for the exactly solvable case $W=x^2/2$: direct substitution gives $-\phi_0$, not $0$, so the seed condition asserted in Section IV.A fails as written; checking whether the recursively defined seeds (23) are instead zero modes of the intermediate Hamiltonian $H^{(s+1)}$ would settle whether the eigenstate formulas (25) and the solution mapping (27) hold.

Watch

Extended reading notes

Core claim

The central discovery is that the same spectrum can be reinstated after deletion, yielding a parametric family of isospectral Fokker-Planck equations. Starting from $H=A^{(0)\dagger}A^{(0)}$ with ground state $\phi_0$ and drift $D^{(0)}=2(\ln|\phi_0|)'$, the n-step Darboux-Crum transform deletes $\phi_0,\ldots,\phi_{n-1}$ and yields a partner with drift $D^{(n)}=2(\ln|\phi_n^{(n)}|)'$; the reverse Darboux-Crum transform, using virtual seed states $\Phi_s(\lambda_s,\ldots,\lambda_{n-1})$ defined recursively in Eq. (23), reinstates those n levels. The resulting Hamiltonian $\hat H^{(0)}$ has exactly the spectrum $\{\epsilon_k\}$ of $H$, its eigenstates are the explicit expressions in Eq. (25), and the drift of the isospectral Fokker-Planck equation is $D^{(n)}=2(\ln|\Phi_0^{-1}(\lambda_0,\ldots,\lambda_{n-1})|)'$. For any solution $P$ of the original equation, Eq. (27) gives a corresponding solution of the deformed equation. In the fractional case (28), the same construction works with each factor $\exp(-\epsilon t)$ replaced by the Mittag-Leffler function $E_\alpha(-\epsilon t^\alpha)$, and the solution correspondence is Eq. (35); the paper notes that the compact operator form (14) no longer holds because the Mittag-Leffler function lacks the exponential addition rule.

Load-bearing premise

The whole construction depends on each seed function $\Phi_s$ generated by Eq. (23) being a genuine zero-energy starting point for the reverse Darboux step, and the paper's check of the first seed in Section IV.A appears to leave an unaccounted-for term $-\phi_0$.

Editorial extensions

If this is right

For any Fokker-Planck equation of the form (1) with a normalizable ground state, the construction yields an n-parameter family of Fokker-Planck equations with identical eigenvalues $\{\epsilon_k\}$; each parameter $\lambda_s$ is chosen outside a forbidden interval so the reinstated state is normalizable.
Given any solution $P$ of the original equation, Eq. (27) immediately produces a solution of the isospectral partner, so every known exactly solvable Fokker-Planck equation spawns families of exactly solvable Fokker-Planck equations without further solving.
The fractional Fokker-Planck equation (28) admits the same delete-and-reinstate construction with drift $D^{(n)}=2(\ln|\Phi_0^{-1}|)'$; solutions of the fractional pair are related by Eq. (35), with temporal factors $E_\alpha(-\epsilon_s t^\alpha)$ replacing $\exp(-\epsilon_s t)$.
The compact operator identity (14) does not carry over to the fractional case because the Mittag-Leffler function lacks the exponential addition formula; only the explicit-sum version (35) survives.
For the thermal-potential approach to black holes, multi-parameter isospectral Fokker-Planck equations can be generated, and the paper argues that whether the transformed thermal potentials correspond to actual black hole systems requires further study.

Reading between the lines

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

Beyond the paper, the same delete-and-reinstate scheme should work with any n distinct eigenstates, not necessarily the lowest n, so the construction generates even more isospectral Fokker-Planck families than Section VI explicitly highlights.
Beyond the paper, because the mapping (27) is algebraic in the eigenfunctions and only the exponential time factors are replaced by Mittag-Leffler functions, the same construction should apply to other relaxation kernels that enter the spectral sum linearly and do not mix eigenvalues.
Beyond the paper, applying Eq. (27) to the exactly solvable case $W=x^2/2$ would produce explicit multi-parameter closed-form solutions that could be checked against numerical integration of the deformed equation, providing a direct test of the eigenstate formulas.
Beyond the paper, for the black-hole thermal-potential application, the parameters $\lambda_i$ would need a physical reading; a concrete test is whether the drift of the isospectral equation defines a temperature that satisfies the same thermodynamic relation as some known black-hole family for particular parameter values.

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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

Colleague,

The headline: this is a modest but legitimate generalization of the one-parameter isospectral FPE construction to n parameters, with explicit eigenstates written out, plus a fractional extension. The underlying machinery is standard Darboux-Crum and SUSY quantum mechanics, and the paper is honest about that. It cites relevant literature, including its own earlier work, and the black-hole comment in the last section is fair and cautious rather than a sales pitch.

What is actually new: the explicit all-eigenstate formulas (25)–(27) for the n-parameter partner, and the observation that the same construction carries over to a fractional FPE with Mittag-Leffler time factors. The mathematics is not deep, but it is useful to have the formulas written down.

The soft spots are concentrated in the written justification, and they are real. In Section IV.A the paper claims that Φ0 = (I0+λ0)/ϕ0 is the general solution of A^(0)†Φ = 0. Direct computation with A^(0)† = −∂x − (ln|ϕ0|)′ gives A†Φ0 = −ϕ0, not zero. What is actually true is that H^(1)Φ0 = 0. That is a demonstrable error in the seed-function step. Second, Eq. (27) (and its fractional analogue Eq. (35)) uses Φ0 as the prefactor in P^(n). Since the new drift is D = 2(ln|Φ0^{-1}|)′, the solution prefactor should be Φ0^{-1}, not Φ0; with the printed Φ0 the ground-state term collapses to a constant, which is not a solution. There are also small typos in Eq. (23) and elsewhere.

These issues are local and almost certainly fixable; the construction itself is standard and the explicit eigenstate formulas follow once the seed functions are set up correctly. Still, as printed the central solution map is not established, so a referee would need to ask for a corrected derivation.

Who benefits: researchers working on solvable FPEs and SUSY QM, who will want the n-parameter formulas in a corrected form. It is a small contribution, but a real one.

My recommendation: send it to peer review, but insist on the two corrections above before acceptance. I would not cite it in its current form.

Referee Report

2 major / 7 minor

Summary. The paper constructs an n-parameter isospectral family of Fokker-Planck equations from a given FPE with constant diffusion and time-independent drift. The construction uses a Darboux-Crum deletion of the lowest n eigenstates of the associated Schr\"odinger operator, followed by inverse Darboux transformations that reinstate the energy levels using virtual states of the form \Phi_s = (I_s + \lambda_s)/\phi_s^{(s)}. Explicit formulas are given for all eigenstates of the isospectral Hamiltonian (Eq. (25)) and for the corresponding solutions of the deformed FPE (Eq. (27)), with a fractional generalization based on Mittag-Leffler functions (Eq. (35)). The paper also comments on a recent application of one-parameter isospectral FPEs to black-hole thermal potentials.

Significance. If the construction is repaired, the paper provides a useful explicit multi-parameter generalization of the known one-parameter isospectral FPE transformations, going beyond the usual ground-state-only statements in the SUSY literature by displaying all eigenstates and a solution map. The method is purely algebraic, relies on standard Darboux-Crum/SUSY machinery, and involves no fitting or numerical optimization; the parameters \lambda_s are genuinely free. The fractional extension, though straightforward, is a convenient addition. However, two load-bearing errors in the current version—a false identity in the base case of the virtual-state construction and an incorrect prefactor in the central solution formula—prevent the claims from being accepted as printed.

major comments (2)

[IV.A (Eq. (15))] The paper claims that \Phi_0 = (I_0 + \lambda_0)/\phi_0 satisfies A^{(0)\dagger}\Phi_0 = 0. Direct computation with A^{(0)\dagger} = -\partial_x - (\ln|\phi_0|)' gives A^{(0)\dagger}\Phi_0 = -\phi_0, not zero. The correct statement is H^{(1)}\Phi_0 = A^{(0)}A^{(0)\dagger}\Phi_0 = 0, i.e., \Phi_0 is a zero mode of the Darboux partner Hamiltonian, not of A^{(0)\dagger}. Since this identity is the only justification given for \Phi_0 as the virtual seed in Eqs. (18), (21)\textendash(23), the derivation of the multi-parameter eigenstates is not valid as written. The error is local and can be repaired by replacing the false identity with the correct zero-mode condition.
[Eq. (27) and Eq. (35)] The proposed solution of the isospectral FPE has prefactor \Phi_0. For the stated drift D^{(n)} = 2(\ln|\Phi_0^{-1}|)', the stationary distribution is \Phi_0^{-2}, and the eigenfunction expansion must be proportional to \Phi_0^{-1} times the bracket. In particular, the s = 0 term in the bracket is \hat{\phi}^{(0)}_0 = \Phi_0^{-1}, so the printed prefactor \Phi_0 yields a constant term that does not satisfy the FPE. The prefactor in Eq. (27) and in its fractional counterpart Eq. (35) should be \Phi_0^{-1}.

minor comments (7)

[Eq. (10)] The factorization should be H^{(1)} = A^{(1)\dagger}A^{(1)} + \epsilon_1; the text writes A^{(1)\dagger}A^{(1)\dagger}, and the second operator on the line is printed as A^{(0)\dagger} instead of A^{(1)\dagger}.
[Eq. (12)] The line 'A^{(2)}\phi^{(2)}_2 = 0' appears to be a copy error; for the general n-step case it should be A^{(n)}\phi^{(n)}_n = 0.
[Eq. (18)] The expression 'B^\dagger_0 A0)\phi_k' should read B^\dagger_0 A^{(0)}\phi_k.
[Eq. (34)] The Mittag-Leffler argument should be -(\epsilon_k - \epsilon_n)t^\alpha, matching Eq. (33); as printed it is dimensionally inconsistent.
[IV.A] The phrase 'general solution satisfying A^{(0)\dagger}\Phi(\lambda_0)=0' is the origin of the error in major comment 1; the correct statement is that \Phi_0 is the general solution (up to a non-normalizable homogeneous term) of H^{(1)}\Phi = 0.
[IV] There is a typo 'transfroamtion' that should be 'transformation'.
[VI] There is a typo 'isopectral' that should be 'isospectral'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the multi-parameter isospectral construction is a self-contained algebraic application of standard Darboux-Crum and SUSY results, with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's central claim—that the FPE with drift D^(n)=2(ln|Phi0^{-1}(lambda_0,...,lambda_{n-1})|)' is the n-parameter isospectral partner of Eq. (1)—is a construction, not a prediction. Given an original FPE, the author deletes n lowest eigenstates by the Darboux-Crum transformation (Eq. (13), from Crum's theorem, an external result), then reinstates them via inverse Darboux transformations using virtual states Phi_s=(I_s+lambda_s)/phi_s^(s). The new Hamiltonian is defined as BhatH^(0)=B0^dagger B0, and the partner FPE is defined by the drift that makes Phi0^{-1} its ground state; the isospectrality follows from the SUSY intertwining relations, not from any assumption of the conclusion. The solution map (27) is the standard eigenfunction expansion in the new basis; it is not a fit to a target output. The author's own prior works [16,17] are cited only as background on SUSY treatments of FPE, not as the justification for the multi-parameter construction, which is built on the external references [6,7,9,11]. There are no data fitted, no parameter tuned to reproduce a target quantity, and no uniqueness claim imported from a self-citation. The reviewer-flagged issues—the seed condition A^dagger Phi0=0 being misstated (direct calculation gives A^dagger Phi0=-phi0) and the prefactor in (27) being Phi0 rather than Phi0^{-1}—are internal mathematical errors or typos that do not make the argument circular; they are correctness risks, not instances of the derivation reducing to its own inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The construction introduces no new physical entities. The lambda_s parameters are free family labels, not fitted constants. The main external input is the standard SUSY/Darboux-Crum machinery plus the domain assumption that a normalizable ground state exists. The false seed-function statement in Section IV.A is a derivation error, not an invented entity.

free parameters (1)

lambda_s (s=0,...,n-1) = arbitrary real parameters outside [-I_s(c2),0]
Labels of the isospectral family. They are chosen by hand, not fitted to data, and they control normalizability of the reinstated states. The construction works for any allowed choice.

assumptions (5)

standard math The Darboux-Crum theorem and the Wronskian formula (13) correctly map eigenstates of H to eigenstates of H_n.
Invoked in Section III to delete the lowest n states. The paper cites [7,9] and uses the result without proof.
standard math Each B_s intertwining operator maps eigenstates of the intermediate Hamiltonian to eigenstates of the isospectral Hamiltonian and preserves the spectrum.
This is the core of Sections IV.A-C. The paper cites [6,11] rather than proving the intertwining property, and the explicit excited-state formulas (18), (22), and (25) depend on it.
domain assumption The original Fokker-Planck equation has a normalizable zero-energy ground state phi_0 and a discrete spectrum with 0 <= epsilon_0 < epsilon_1 < ...
Stated in Section II: 'we consider only the case when phi_0 is an eigenstate.' The construction deletes and reinstates the n lowest states based on this spectral structure.
domain assumption The parameters lambda_s must lie outside [-I_s(c2),0] for the seed functions Phi_s^{-1} to be normalizable.
Stated in Eq. (23) and Section IV.A. The isospectral family is restricted to these values to ensure L2 ground states.
domain assumption For the fractional Fokker-Planck equation, the separation P = e^{-W} T(t) phi(x) is valid and T(t) = E_alpha(-epsilon t^alpha) solves the temporal fractional equation.
Section V relies on [22,23]. The Mittag-Leffler function replaces the exponential only for this class of fractional FPE, and the paper notes that the factorization property exp(x+y) is lost.

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Cite this review

Pith. "Pith review of Multi-parameter isospectral Fokker-Planck equations." pith.science (2026). https://pith.science/paper/XHM6FJHB

@misc{pith2026250612939,
  author       = {Pith},
  title        = {Pith review of: Multi-parameter isospectral Fokker-Planck equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHM6FJHB}},
  note         = {Machine review of arXiv:2506.12939}
}

read the original abstract

From a given Fokker-Planck equation, a multi-parameter deformed partner Fokker-Planck equation is constructed. This is done by first deleting a set of eigenstates of the original FPE by the multi-step Darboux-Crum transformation, and then reinstating the eigen-energy levels by the reverse Darboux-Crum transformation. Extension to fractional Fokker-Planck equation is briefly discussed. A recent study of the one-parameter isospectral FPE applied to black hole in the thermal potential approach is commented.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Risken, The Fokker-Planck Equation, 2nd

H. Risken, The Fokker-Planck Equation, 2nd. ed., Springer-Verlag, Berlin, 1996

work page 1996

[2] [2]

Sau Fa Kwok, Langevin and Fokker-Panck equations and their generalizations, World Scien- tific, Singapore, 2018

work page 2018

[3] [3]

Ho and R

C.-L. Ho and R. Sasaki, Quasi-exactly solvable Fokker-Planck equations, Ann. Phys. 323 (2008) 883

work page 2008

[4] [4]

Ho, Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials, Ann

C.-L. Ho, Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials, Ann. Phys. 326 (2011) 797

work page 2011

[5] [5]

Chou and C.-L

C.-I. Chou and C.-L. Ho, Rayleigh and Jacobi processes and exceptional orthogonal polyno- mials, Int. J. Mod. Phys. B 27 (2013) 1350135

work page 2013

[6] [6]

Cooper, A

F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry in quantum mechanics, World Sci- 13 entific, Singapore, 2001

work page 2001

[7] [7]

Sasaki, Exactly solvable quantum mechanics, The Universe, Vol 2 (2014) 2

R. Sasaki, Exactly solvable quantum mechanics, The Universe, Vol 2 (2014) 2

work page 2014

[8] [8]

Darboux, Sur une proposition relative aux ´ equation lin´ eaires, Compte Rendu Acad

G. Darboux, Sur une proposition relative aux ´ equation lin´ eaires, Compte Rendu Acad. Sci. Paris 94 (1882) 1456

work page