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arxiv: 2604.15908 · v1 · submitted 2026-04-17 · ❄️ cond-mat.stat-mech · math.PR

On the role of the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with N=2 explicit levels

Cecile Monthus This is my paper

Pith reviewed 2026-05-10 07:19 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR
keywords Markov processesquasi-exactly-solvabledetailed balanceFokker-Planck generatorsslowest observablerelaxation ratesteady stateone-dimensional dynamics
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0 comments X p. Extension

The pith

Taking the slowest observable as the central object simplifies the construction of quasi-exactly-solvable Markov generators with two explicit levels.

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

The paper revisits quasi-exactly-solvable quantum Hamiltonians by mapping them to one-dimensional Markov processes that obey detailed balance, where the generators relate through similarity transformations. It shows that the lowest eigenvalue vanishes and corresponds to the steady-state probability, while the first positive eigenvalue sets the relaxation rate and is carried by the slowest observable L1(x). The central argument is that starting from this observable makes the full construction of models with exactly two explicit levels more intuitive and technically simpler than working directly with the quantum eigenfunctions. This holds for both continuous Fokker-Planck generators and discrete Markov jump generators on lattices.

Core claim

In one-dimensional Markov processes satisfying detailed balance, the construction of quasi-exactly-solvable generators with N=2 explicit levels becomes more intuitive and technically simpler when the slowest observable L1(x), defined as the ratio of the two lowest eigenfunctions, is taken as the central object from which the steady-state distribution and the generator itself are reconstructed.

What carries the argument

The slowest observable L1(x) = Φ1(x)/Φ0(x), which encodes the relaxation rate E1 and serves as the starting point to recover the steady-state probability P*(x) and the full Markov generator via the similarity to the quantum Hamiltonian.

If this is right

  • Choosing a suitable functional form for L1(x) directly yields valid Fokker-Planck generators in continuous space with exactly two explicit levels.
  • The same L1(x)-centered procedure produces quasi-exactly-solvable Markov jump generators on discrete lattices.
  • The steady-state probability P*(x) and the generator follow algebraically once L1(x) and the detailed-balance condition are specified.
  • The relaxation dynamics is exactly governed by the single rate E1 tied to L1(x) while all higher modes remain implicit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Selecting L1(x) with additional symmetry properties could produce families of models that admit further exact results beyond the first two levels.
  • The method may extend to driven or time-dependent Markov processes whenever a dominant slow observable can be identified a priori.
  • Numerical schemes that preserve the exact slow dynamics while approximating faster modes could be designed by fixing L1(x) from data.
  • Long-time large-deviation statistics might be simplified because the slowest observable dominates the exponential relaxation.

Load-bearing premise

That designating the slowest observable L1(x) as the central object genuinely makes the construction more intuitive and technically simpler than standard quantum approaches.

What would settle it

An explicit side-by-side algebraic construction of one concrete quasi-exactly-solvable model, such as a quartic or double-well case, using both the conventional quantum route and the L1(x)-centered Markov route, to compare the number of steps and intermediate expressions required.

read the original abstract

The construction of Quasi-Exactly-Solvable quantum Hamiltonians where only the two first eigenstates $\Phi_0(x)$ and $\Phi_1(x)$ of energies $E_0$ and $E_1$ are explicit is revisited from the point of view of one-dimensional Markov processes satisfying detailed-balance, whose generators are related to quantum Hamiltonians via similarity transformations. Here the lowest energy vanishes $E_0=0$ and is associated the conservation of probability and to the steady state $P_*(x)$, while $E_1>0$ is the rate that governs the exponential relaxation towards the steady-state, and is associated to the slowest observable $L_1(x)$ that corresponds to the ratio $ \frac{\Phi_1(x) }{\Phi_0(x)}$ of the two quantum eigenstates. Our main conclusion is that the Markov perspective leads to interesting re-interpretations and that the construction of quasi-exactly-solvable models with $N=2$ explicit levels is more intuitive and technically simpler if one takes the slowest observable $L_1(x)$ as the central object from which all the other properties can be reconstructed. This general approach is then applied to Fokker-Planck generators in continuous space and to Markov jump generators on the lattice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript revisits the construction of quasi-exactly-solvable (QES) quantum Hamiltonians with only the ground and first excited states explicit, reframing it through one-dimensional Markov processes that obey detailed balance. The generators are linked to quantum Hamiltonians by similarity transformations, with E0=0 tied to probability conservation and the steady state P*(x), while E1>0 governs relaxation and is associated with the slowest observable L1(x) ≡ Φ1(x)/Φ0(x). The central claim is that centering the construction on L1(x) yields more intuitive re-interpretations and technically simpler derivations of the steady state and generator for N=2 cases; explicit constructions are given for both continuous Fokker-Planck operators and discrete lattice jump processes, with the quantum similarity transformation recovered automatically.

Significance. If the explicit constructions hold without hidden parameter fitting, the Markov-centric route supplies a useful change of perspective that may streamline the generation of solvable stochastic models in statistical mechanics. The automatic recovery of the quantum mapping and the provision of both continuous and discrete examples add concrete value, though the asserted gain in simplicity remains partly subjective and would be strengthened by side-by-side comparisons.

major comments (2)
  1. [§3] §3 (Fokker-Planck construction): the derivation of the drift and diffusion coefficients from a chosen L1(x) via the detailed-balance relation is presented algebraically, but the manuscript does not include a direct, step-by-step comparison with the conventional quantum-Hamiltonian construction for the same L1(x) example; without this, the claim of technical simplicity remains qualitative rather than demonstrated.
  2. [§4] §4 (lattice jump process): the reconstruction of the transition rates from L1(x) and P*(x) is shown to satisfy the master equation, yet the text does not verify that the resulting generator is free of post-hoc adjustments for arbitrary choices of L1(x) on finite lattices; an explicit counter-example or general proof would strengthen the generality asserted in the abstract.
minor comments (3)
  1. [Abstract] Abstract, line 8: the phrasing 'the lowest energy vanishes E0=0 and is associated the conservation of probability' contains a grammatical omission ('associated to' or 'associated with').
  2. [Introduction] Notation: the symbol L1(x) is introduced in the abstract but its explicit definition as Φ1(x)/Φ0(x) appears only later; a brief parenthetical reminder in the first use of the main text would improve readability.
  3. [Figures] Figure captions: the parameters chosen for the plotted examples (e.g., specific forms of L1(x)) are not stated in the captions, making it harder to reproduce the figures from the text alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (Fokker-Planck construction): the derivation of the drift and diffusion coefficients from a chosen L1(x) via the detailed-balance relation is presented algebraically, but the manuscript does not include a direct, step-by-step comparison with the conventional quantum-Hamiltonian construction for the same L1(x) example; without this, the claim of technical simplicity remains qualitative rather than demonstrated.

    Authors: We agree that an explicit side-by-side comparison would render the claimed technical advantage more concrete rather than qualitative. In the revised manuscript we have inserted a new subsection (3.3) that takes the same explicit L1(x) example used in §3.2 and derives the drift and diffusion coefficients first via the Markov route (using only the detailed-balance relation and the definition of L1) and then via the conventional quantum route (constructing the potential from the ratio of eigenfunctions and solving the resulting Schrödinger equation). The comparison shows that the Markov construction bypasses the intermediate step of determining the potential and directly yields the generator coefficients, confirming the simplification for N=2 cases. revision: yes

  2. Referee: [§4] §4 (lattice jump process): the reconstruction of the transition rates from L1(x) and P*(x) is shown to satisfy the master equation, yet the text does not verify that the resulting generator is free of post-hoc adjustments for arbitrary choices of L1(x) on finite lattices; an explicit counter-example or general proof would strengthen the generality asserted in the abstract.

    Authors: The construction in §4 is algebraic and by design contains no post-hoc adjustments: once L1(x) and P*(x) are chosen (subject only to the requirement that the resulting rates remain non-negative), the transition rates are uniquely fixed by the two equations given in the text, and the master equation together with detailed balance are satisfied identically. To make this generality explicit we have added, in the revised §4, a short general argument showing that the rate formulas automatically produce a valid stochastic generator for any admissible L1 on a finite lattice, together with a concrete numerical example on a three-site chain where an arbitrary polynomial L1 is inserted and the rates are computed without further tuning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reinterpretation is self-contained algebraic reconstruction

full rationale

The paper re-expresses the standard construction of N=2 quasi-exactly-solvable models by designating the slowest observable L1(x) = Φ1(x)/Φ0(x) as the starting point, then recovers the steady-state P*(x), the Markov generator, and the similarity-transformed quantum Hamiltonian via the detailed-balance relation. These steps are direct algebraic consequences of the known equivalence between detailed-balance Markov generators and quantum Hamiltonians; they do not define L1(x) in terms of the quantities it is used to predict, nor do they fit parameters to data and relabel the fit as a prediction. No self-citation chain is invoked as a uniqueness theorem, and the claim of greater intuitiveness is presented as a change of perspective rather than a load-bearing mathematical result. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard domain assumptions of one-dimensional Markov processes obeying detailed balance to enable the similarity transformation to quantum Hamiltonians; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Markov processes satisfy detailed balance
    This property is required for the similarity transformation relating the Markov generator to a quantum Hamiltonian with real spectrum.
  • domain assumption The processes are one-dimensional
    The construction and reconstruction from the slowest observable are developed specifically for 1D continuous and discrete cases.

pith-pipeline@v0.9.0 · 5538 in / 1340 out tokens · 42210 ms · 2026-05-10T07:19:46.503276+00:00 · methodology

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Reference graph

Works this paper leans on

93 extracted references · 2 canonical work pages

  1. [1]

    Continuity Equation for the probabilityP t(x)involving the divergence of the currentJ t(x) The dynamics of the probabilityP t(x) is governed by the continuity equation that involves the divergence of the currentJ t(x) ∂tPt(x) =−∇J t(x) (1) where the explicit form of the operator∇depends on the continuous or discrete nature of the space ∇ ≡ ∂ ∂x derivative...

  2. [2]

    5, while its adjoint reads J† =−e U(x) ∇†e−UI(x) =e U(x) ∇e−UI(x) (8) 3

    Parametrization of the current operatorJproducing the currentJ t(x) =JP t(x) The currentJ t(x) can be computed from the probabilityP t(x) via some current operatorJ Jt(x) =JP t(x) (4) We will focus on models where the steady stateP ∗(x) that will be parametrized by the potentialU(x) and some normalizationZ P ∗(x) = e−U(x) Z (5) is associated to a vanishin...

  3. [3]

    7 and its adjointJ † of Eq

    Discussion In summary, for the general analysis of the present section, it is convenient to parametrize the one-dimensional reversible Markov models by the two potentialsU(x) andU I(x), whose exponentials appear around the operator∇ in the current operatorJof Eq. 7 and its adjointJ † of Eq. 8. The link with more standard parametrizations in terms of two o...

  4. [4]

    Similarity transformationG=−e − U(x) 2 He U(x) 2 towards an hermitian supersymmetric quantum HamiltonianH=Q †Q The factorized form of Eq. 10 for the Markov generatorGsuggests the rewriting G=−e − U(x) 2 −e U(x) 2 ∇e− UI (x) 2 e− UI (x) 2 ∇e U(x) 2 e U(x) 2 ≡ −e − U(x) 2 Q†Qe U(x) 2 ≡ −e − U(x) 2 He U(x) 2 (14) that makes obvious the well-known similarity ...

  5. [5]

    5 Φ0(x) = p P ∗(x) = e− U(x) 2 √ Z (21) The similarity transformation of Eq

    Links between the spectral properties of the Markov generatorGand of the quantum HamiltonianH=Q †Q The spectral decomposition of the evolution operatore −Ht associated to the quantum HamiltonianH e−Ht = +∞X n=0 e−tEn |Φn⟩⟨Φn|(18) involves its real eigenvaluesE n, that will be assumed to be all discrete for the present general discussion to simplify the no...

  6. [6]

    Discussion : why it is simpler to focus on the left eigenvectorsL n(x)only To analyze the spectral properties of the Markov generatorG, it is actually simpler to focus on the left eigenvectors Ln(x) only, since they satisfy the simpler eigenvalue Eq. 24, starting with the trivial eigenvector forn= 0 associated to the conservation of probability Ln=0(x) = ...

  7. [7]

    Links between the spectral properties of the quantum HamiltonianH=Q †Qand its partner ˘H=QQ † The main output of supersymmetric quantum mechanics (see the reviews [1–5]) is that the spectral decomposition of the evolution operatore − ˘Ht associated to the supersymmetric partner ˘H=QQ † e− ˘Ht = +∞X n=1 e−tEn |˘Φn⟩⟨˘Φn|(28) involves the non-vanishing eigen...

  8. [8]

    Rewriting the quantum partner ˘H=QQ † as ˘H=E 1 + Q[1] † Q[1] whereQ [1] annihilates the ground state ˘Φ1(x) As explained around Eq. 28, the quantum partner ˘H=QQ † has the positive ground-state energyE 1 >0, so that it is useful to introduce the new potentialU [1](x) that parametrizes the corresponding positive ground state ˘Φ1(x) ˘Φ1(x) = e− U [1] (x) 2...

  9. [9]

    38 is related to the quantum partner ˘H=QQ † of Eq

    Similarity transformation between the partners ˘G=−J∇and ˘H=QQ † with consequences for their spectral properties The partner ˘G=−J∇introduced in Eq. 38 is related to the quantum partner ˘H=QQ † of Eq. 27 via the similarity transformation ˘G=e −UI(x)∇eU(x) ∇ =−e − UI (x) 2 −e− UI (x) 2 ∇eU(x) ∇e− UI (x) 2 e UI (x) 2 =−e − UI (x) 2 ˘He UI (x) 2 (40) 7 This ...

  10. [10]

    Links between the spectral properties of the generatorG=−∇Jand of its partner ˘G=−J∇ Let us translate the two relations of Eq. 30 between the quantum eigenstates Φ n(x) and ˘Φn(x) by plugging their expressions in terms of the left eigenvectorsL n(x) and ˘Ln(x) using Eqs 23 and 42 1 cn e− UI (x) 2 ˘Ln(x) = h e− UI (x) 2 ∇e U(x) 2 i e− U(x) 2√ Z Ln(x) √En e...

  11. [11]

    Doob-transformation from the partner ˘Gtowards the Markov generatorG [1] with steady state P [1] ∗ (x) = ˘Φ2 1(x) = ˘L1(x) ˘R1(x) In the field of Markov processes, when an operator like the partner ˘Gappears with a non-vanishing ground state energyE 1 >0 associated to the left eigenvector ˘L1(x) and to the right eigenvector ˘R1(x), it is often useful to c...

  12. [12]

    Link between the new Markov generatorG [1] and the new quantum HamiltonianH [1] with vanishing ground state energy The ground state Φ [1] 0 (x) ofH [1] that coincides with the groundstate ˘Φ1(x) of ˘Hcan be computed via the relation of Eq. 30 forn= 1 that involves the first-excited-state Φ 1(x) = Φ0(x)L1(x) = e− U(x) 2√ Z L1(x) of the initial Hamiltonian ...

  13. [13]

    34 is constructed via the supersymmetric recursionH [1] = ˘H−E 1 of Eq

    Discussion In summary, we have described the equivalence between two points of view to find the two new potentialsU[1](x)and U[1] I (x) in terms of the two initial potentialsU(x) andU I(x) : (i) In the quantum perspective, the new HamiltonianH [1] =−e U [1] (x) 2 ∇e−U [1] I (x)∇e U [1] (x) 2 of Eq. 34 is constructed via the supersymmetric recursionH [1] =...

  14. [14]

    61 withO(x) =e U(x) to the current operator of Eq

    Link between the two parametrizations of the current operatorJ=−e −UI(x) ∂ ∂x eU(x) =F(x)−D(x) ∂ ∂x The application of the rule of Eq. 61 withO(x) =e U(x) to the current operator of Eq. 7 J=−e −UI(x) ∂ ∂x eU(x) =−e −UI(x) U ′(x)eU(x) +e U(x) ∂ ∂x =−e U(x)−U I(x) U ′(x) + ∂ ∂x (62) yields that the identification with the standard parametrization of the cur...

  15. [15]

    61 withO(x) =e −UI(x) to the adjoint of Eq

    Explicit forms of the adjointsG † and ˘G† with the corresponding eigenvalue equations forL n(x)and ˘Ln(x) The application of the rule of Eq. 61 withO(x) =e −UI(x) to the adjoint of Eq. 8 J† =e U(x) ∂ ∂x e−UI(x) =e U(x) e−UI(x) ∂ ∂x −U ′ I(x)e−UI(x) =e U(x)−U I(x) ∂ ∂x −U ′ I(x) =D(x) ∂ ∂x −U ′ I(x) ≡D(x) ∂ ∂x +F I(x) (67) shows that it is also useful to i...

  16. [16]

    65, the operatorsQof Eq

    Explicit form of the quantum HamiltonianH=Q †Q UsingU I(x) =U(x)−lnD(x) of Eq. 65, the operatorsQof Eq. 16 andQ † of Eq. 17 become the first-order differential operators Q=e − UI (x) 2 ∂ ∂x e U(x) 2 = p D(x)e− U(x) 2 ∂ ∂x e U(x) 2 = p D(x) ∂ ∂x + U ′(x) 2 Q† =−e U(x) 2 ∂ ∂x e− UI (x) 2 =−e U(x) 2 ∂ ∂x e− U(x) 2 p D(x) = − ∂ ∂x + U ′(x) 2 p D(x) (74) Then ...

  17. [17]

    27 ˘H=−e − UI (x) 2 ∇eU(x) ∇e− UI (x) 2 =QQ † =− ∂ ∂x D(x) ∂ ∂x + ˘V(x) (77) 12 involves the same kinetic term ∂ ∂x D(x) ∂ ∂x as the HamiltonianHof Eq

    Explicit form of the quantum partner ˘H=QQ † The supersymmetric partner ˘H=QQ † of Eq. 27 ˘H=−e − UI (x) 2 ∇eU(x) ∇e− UI (x) 2 =QQ † =− ∂ ∂x D(x) ∂ ∂x + ˘V(x) (77) 12 involves the same kinetic term ∂ ∂x D(x) ∂ ∂x as the HamiltonianHof Eq. 75, while the partner scalar potential ˘V(x) given by ˘V(x)≡D(x) [U ′(x)]2 4 +D(x) U ′′(x) 2 + [D′(x)]2 4D(x) − D′′(x)...

  18. [18]

    77 into the new HamiltonianH [1] of Eq

    Construction of the new HamiltonianH [1] ≡ Q[1] † Q[1] = ˘H−E 1 =QQ † −E 1 Plugging the explicit form of ˘Hof Eq. 77 into the new HamiltonianH [1] of Eq. 36 H[1] = ˘H−E 1 =− ∂ ∂x D(x) ∂ ∂x + ˘V(x)−E 1 =− ∂ ∂x D(x) ∂ ∂x +V [1](x) (79) yields thatH [1] involves the same kinetic term ∂ ∂x D(x) ∂ ∂x with the same diffusion coefficientD(x), while the new scala...

  19. [19]

    64 and the Ito forceF I(x) =F(x) +D ′(x) of Eq

    Reminder on the Stratonovich forceF S(x) =F(x) + D′(x) 2 =F I(x)− D′(x) 2 Besides the Fokker-Planck forceF(x) =−D(x)U ′(x) of Eq. 64 and the Ito forceF I(x) =F(x) +D ′(x) of Eq. 68, it is often useful to introduce also the Stratonovich forceF S(x) related to both previous forces via FS(x) =F(x) + D′(x) 2 =F I(x)− D′(x) 2 (115) with the corresponding poten...

  20. [20]

    Transformation rules for the various properties of the Fokker-Planck dynamics for a change of variablesx→˚x •For all the observablesO(x), and thus in particular for the left eigenvectorsL n(x), the change of variables reduces to a change in the argument ˚Ln(˚x) =Ln(x) x=x(˚x) (120) •The corresponding change of the adjoint of Eq. 119 written in terms of th...

  21. [21]

    Specific properties of the translation operatorse b∂ in discrete space The translation operatorse b∂ bybacts on any functionO(x) via the Taylor formula eb∂O(x) = +∞X p=0 bp p! ∂ ∂x p O(x) =O(x+b) (A2) The application to product of two functionsO(x) and Ω(x) eb∂O(x)Ω(x) =O(x+b)Ω(x+b) (A3) yields the following rule at the operator level for the exchange of ...

  22. [22]

    GeneratorGas a tridiagonal Markov matrix parametrized by the two potentialsU(.)andU I .+ 1 2 Using the finite-difference operator of Eq

    Explicit forms of the generator G and its partner ˘G as tridiagonal matrices a. GeneratorGas a tridiagonal Markov matrix parametrized by the two potentialsU(.)andU I .+ 1 2 Using the finite-difference operator of Eq. A1, the continuity Eq. 1 becomes ∂tPt(x) =−∇J t(x) =− e ∂ 2 −e − ∂ 2 Jt(x) =J t x− 1 2 −J t x+ 1 2 (A6) while the current of Eq. 4 with 7 re...

  23. [23]

    ˘Ln x+ 1 2 −e U(x)−U I(x− 1

  24. [24]

    ˘Ln x− 1 2 ≡G †(x, x+ 1) ˘Ln x+ 1 2 −G †(x, x−1) ˘Ln x− 1 2 (A12) when one introduces the left eigenvectors ˘Ln(x) of the partner ˘Gwritten in the next section. b. Explicit form of the partner ˘Gas a tridiagonal matrix The dynamics of the currentJ t x+ 1 2 of Eq. A8 obtained using the continuity Eq. A6 ∂tJt x+ 1 2 =−e −UI(x+ 1 2)+U(x+1) ∂tPt(x+ 1) +e −UI(...

  25. [25]

    Explicit form of the quantum HamiltonianH=Q †Qas a tridiagonal matrix The similarity transformation of Eq

    Explicit forms of the quantum Hamiltonians H=Q †Q and ˘H=QQ † and H[1] = ˘H−E 1 = Q[1] † Q[1] a. Explicit form of the quantum HamiltonianH=Q †Qas a tridiagonal matrix The similarity transformation of Eq. 14 yields that the HamiltonianHis a symmetric tridiagonal matrix, whose matrix elements can be directly obtained from the matrix elements ofGof Eq. A10 H...

  26. [26]

    =H(x+ 1, x) H(x−1, x) =−e U(x−1) 2 G(x−1, x)e − U(x) 2 =−e U(x)+U(x−1) 2 −UI(x− 1

  27. [27]

    =H(x, x−1) H(x, x) =−G(x, x) =e U(x)−U I(x+ 1

  28. [28]

    (A16) To make the link with the quantum Hamiltonian of Eq. 75 in continuous space where the diffusion coefficientD(x) is the amplitude of the kinetic term, it is useful to define the diffusion coefficientD x+ 1 2 on each bond of the discrete model from the off-diagonal elements of Eq. A16 D x+ 1 2 ≡ −H(x+ 1, x) =e U(x+1)+U(x) 2 −UI(x+ 1

  29. [29]

    A10 between the two sitesxand (x+ 1) in the two directions

    = p G(x+ 1, x)G(x, x+ 1) (A17) where the last expression gives the re-interpretation in terms of the two jump ratesG(x+ 1, x) andG(x, x+ 1) of Eq. A10 between the two sitesxand (x+ 1) in the two directions. This equation can be used to write the replacement UI x+ 1 2 = U(x+ 1) +U(x) 2 −lnD x+ 1 2 (A18) that is the discrete counterpart of the continuous re...

  30. [30]

    50 yields that the matrix elements of G[1] † can be constructed from the matrix elements of ˘G† written in Eq

    Construction of the new Markov matrix G [1] via the Doob-transformation of the partner ˘G The Doob-transformation of Eq. 50 yields that the matrix elements of G[1] † can be constructed from the matrix elements of ˘G† written in Eq. A14 and from the left eigenvector ˘L1(.) via G[1] † x− 1 2 , x+ 1 2 = 1 ˘L1 x− 1 2 ˘G† x− 1 2 , x+ 1 2 ˘L1 x+ 1 2 = ˘L1 x+ 1 ...

  31. [31]

    =G [1] x+ 1 2 , x− 1 2 G[1] † x+ 1 2 , x− 1 2 =e −U [1] I (x)+U [1](x+ 1

  32. [32]

    =G [1] x− 1 2 , x+ 1 2 G[1] † x+ 1 2 , x+ 1 2 =− h e−U [1] I (x)+U [1](x+ 1

  33. [33]

    +e −U [1] I (x+1)+U [1](x+ 1 2)i =G [1] x+ 1 2 , x+ 1 2 (A35) leads to the following discussion : (1) The identification of the two types of off-diagonal elements of G[1] † x± 1 2 , x∓ 1 2 between Eqs A34 and A35 can be rewritten as equations for the same ratio ˘L1(x+ 1 2) ˘L1(x− 1

  34. [34]

    of two consecutive components of the left eigenvector ˘L1(.) ˘L1 x+ 1 2 ˘L1 x− 1 2 = G[1] † x− 1 2 , x+ 1 2 ˘G† x− 1 2 , x+ 1 2 = e−U [1] I (x)+U [1](x− 1 2) e−UI(x+ 1 2)+U(x) ˘L1 x+ 1 2 ˘L1 x− 1 2 = ˘G† x+ 1 2 , x− 1 2 G[1] † x+ 1 2 , x− 1 2 = e−UI(x− 1 2)+U(x) e−U [1] I (x)+U [1](x+ 1

  35. [35]

    (A36) The compatibility between these two equations yields G[1] † x− 1 2 , x+ 1 2 G[1] † x+ 1 2 , x− 1 2 = ˘G† x− 1 2 , x+ 1 2 ˘G† x+ 1 2 , x− 1 2 i.e.e −2U [1] I (x)+U [1](x− 1 2)+U [1](x+ 1

  36. [36]

    =e −UI(x+ 1 2)−UI(x− 1 2)+2U(x) i.e.D [1](x) = ˘D(x) (A37) which is equivalent to Eq. A26 concerning the off-diagonal matrix elements of the quantum Hamiltonians parametrized by the diffusion coefficientsD [1](x) = ˘D(x) The remaining independent equation can be chosen to be the product of the two equations of Eq. A36 ˘L1 x+ 1 2 ˘L1 x− 1 2 !2 = eUI(x+ 1 2...

  37. [37]

    (A38) that corresponds to Eq. 55 of the general analysis ˘L1 x+ 1 2 =e UI(x+ 1 2)−U [1](x+ 1 2) 2 (A39) (2) The identification of the diagonal elements of G[1] † in Eqs A34 and A35 G[1] † x+ 1 2 , x+ 1 2 = ˘G† x+ 1 2 , x+ 1 2 +E 1 i.e.e −U [1] I (x)+U [1](x+ 1

  38. [38]

    +e −U [1] I (x+1)+U [1](x+ 1

  39. [39]

    A28 concerning the diagonal matrix elements of the quantum Hamiltonians

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