Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics

Jean-Marc Sparenberg; Quentin Bozet

REVIEW 3 major objections 4 minor 22 references

Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The two steps of supersymmetric inversion—phase-shift fixing and bound-state addition—collapse into one closed Wronskian formula.

desk verdict A genuinely useful unification for the SUSYQM inversion community, with a rigor gap on the imported theorem and an overreach on the 1S0 claim. read the letter →

arxiv 2508.19022 v1 pith:HILZ2BJW submitted 2025-08-26 nucl-th

Quentin Bozet , Jean-Marc Sparenberg This is my paper · ORCID

classification nucl-th MSC 81U4081Q60 PACS 03.65.Nk03.65.-w21.30.-x

keywords supersymmetricquantummechanicsinversescatteringWronskianformulationconfluentSUSYtransformationsphase-equivalentpotentialsbound-stateadditionneutron-protonCrum-Kreinformula

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

This paper claims that the two steps of the supersymmetric inversion method—first fixing the scattering phase shift with a chain of non-confluent transformations, then adding bound states with a confluent pair—can be collapsed into one closed Wronskian formula. The formula expresses the final potential entirely through solutions of the original potential at the chosen factorization energies, so for simple starting potentials the final interaction is analytic. If correct, it replaces the numerical integral step that previously blocked analytical phase-equivalent bound-state addition, and it yields explicit analytic neutron-proton potentials in the 3S1 and 1S0 waves. A derived identity shows the Wronskian form contains the older integral parametrization, so the family of potentials with a given phase shift and one bound state is spanned exactly.

What carries the argument

The central object is the Wronskian built from factorization solutions of the original Schrödinger equation, with the final entry a generalized eigenfunction of a Jordan chain: u1(r,E) = beta phi0(r,E)_perp + partial_E phi0(r,E). The multi-confluent Wronskian theorem of the paper's reference [2] lets a chain of M non-confluent transformations followed by a confluent pair at the same energy be represented by this single Wronskian; taking -2 times the second logarithmic derivative of the Wronskian gives the final potential. This object does the work of replacing the integral formula (23) that could not be combined with the Crum-Krein formula, and of carrying both phase-shift fixing and bound-s

What would settle it

Compute both sides of identity (33) for a factorization solution that is not a pure exponential—say phi0(r,E)=sinh(kappa r)/r or a Coulomb wave—using arbitrary-precision quadrature; any disagreement beyond round-off refutes the claimed equivalence of the integral and Wronskian formulations, and with it Eq. (31). Alternatively, in the M=0, V0=0 test case, construct V2 from the Wronskian and from the integral formula, solve the Schrödinger equation numerically for both, and compare the phase shift and bound-state energy.

Watch

Extended reading notes

Core claim

The central claim is equation (31): after M non-confluent transformations at energies E1,...,EM followed by a confluent pair at energy E, the final potential is V0(r) - 2 d2/dr2 ln W[phi0(E1),...,phi0(EM), phi0(E), beta phi0(E)_perp + partial_E phi0(E)], with all entries built from solutions of the original potential V0. The last entry is a generalized eigenfunction of a Jordan chain of length two, with beta controlling the asymptotic normalization constant. The paper also establishes identity (33), ∫_r^∞ phi0(t,E)^2 dt = W[phi0(r,E), partial_E phi0(r,E)], which makes the new Wronskian form exactly equivalent to the integral parametrization (23) for phase-equivalent pairs, covering bound-sta

Load-bearing premise

The load-bearing premise is that the previously proved Wronskian formula for multi-confluent supersymmetric chains applies to the chains built in this paper, whose auxiliary functions are non-normalizable; if that formula requires tighter hypotheses on these functions, equation (31) does not follow.

Editorial extensions

If this is right

For any starting potential V0 simple enough, equation (31) gives the final phase-equivalent potential analytically, with no intermediate numerical integrations.
The identity (33) proves the integral and Wronskian parametrizations coincide, so the single parameter beta fully controls the bound-state asymptotic normalization constant in both.
For the 3S1 neutron-proton wave, equation (36) spans the entire family of potentials with the low-energy phase shift and one bound state; the physical deuteron case is recovered by coalescing κ2=κ1 in equation (37).
For the 1S0 wave, equation (39) provides an analytic family of deep potentials with one forbidden bound state that fit the full elastic phase shift, opening a systematic route to spin- and parity-independent nucleon-nucleon potentials.
Adding more bound states only requires appending further pairs of functions (u0,u1) to the Wronskian at the desired energies, following the multi-confluent theorem.

Reading between the lines

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

The imported multi-confluent Wronskian theorem is not re-proved for the non-normalizable generalized solutions used here; a direct proof of Eq. (31) under the paper's boundary conditions would settle whether the formula holds exactly or only for sufficiently regular chains.
The failure of symbolic, automatic-differentiation, and numerical evaluation of the eight-function Wronskian (39) suggests that the closed formula's value is algebraic, while computations may need a reformulation—for example, evaluating the equivalent integral chain stepwise or using a scaled determinant form.
The identity (33) may extend beyond the exponential and hyperbolic examples checked in the paper; testing it with Coulomb or other non-exponential factorization solutions could widen the Wronskian formulation to long-range potentials.
If stable evaluation is found, the unified formula turns the whole fixed-angular-momentum inversion problem into a purely algebraic construction from experimental phase-shift and bound-state parameters, which could make high-precision coupled-channel inversions practical.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: the paper delivers a real, modest unification. Equation (31) is new — it fuses the non-confluent Crum-Krein Wronskian with Schulze-Halberg's confluent-chain result, using Bermudez's generalized eigenfunction, and the 3S1 closed form (36) genuinely covers the whole phase-equivalent family with one bound state at arbitrary energy and ANC. The deuteron limit reproduces the known Eckart potential, and identity (33) checks out in the exponential case. That is a concrete contribution, and the authors credit [1],[2],[15] properly.

The main soft spot is the import of [2]'s theorem. Its hypotheses are never stated, and the chains here run through singular intermediate potentials, so the applicability is asserted rather than shown. I don't think it's wrong — the M=2 case works numerically, and the structure is a standard Darboux chain — but a referee should make the authors spell out what [2] requires and confirm it holds for their cases.

The bigger practical problem is the 1S0 application. Formula (39) is compact and elegant, but the authors admit that neither symbolic, automatic-differentiation, nor numerical evaluation succeeded. That makes the flagship new capability unusable so far, and the abstract's 'complete solution' language overreaches. The 1S0 claim should be softened to 'closed form pending stable evaluation.' Also, identity (33) is cited rather than derived; minor, since it is verified for the relevant exponential case.

Citation pattern is honest. No inflation. The paper is niche but it is a step forward for the SUSYQM inversion community. It deserves a serious referee: the central derivation should be checked against [2]'s hypotheses, and the claims on 1S0 reset. I'd send it out.

Referee Report

3 major / 4 minor

Summary. The paper proposes a unified Wronskian formula (Eq. (31)) that combines the two steps of the supersymmetric inversion scheme: a chain of non-confluent SUSYQM transformations that fixes the scattering phase shift, followed by a confluent pair that adds (or removes/modifies) a bound state without changing the phase shift. The final potential is expressed directly in terms of factorization solutions of the original potential V0, replacing the integral formula (23). The authors also derive the identity (33) relating the integral and Wronskian forms. The formalism is applied to neutron-proton scattering: for the 3S1 channel, an explicit family of phase-equivalent potentials with one bound state of arbitrary energy and ANC is given by Eq. (36), with the deuteron case in Eq. (37); for the 1S0 channel, a formal expression (39) for a deep potential with one forbidden bound state is written down. Numerical verification is reported for the 3S1 case, while the 1S0 expression is stated to be numerically unstable.

Significance. If Eq. (31) is valid under the conditions needed for the singular and non-normalizable solutions used here, it is an elegant and useful unification: it extends the Crum-Krein Wronskian formula to confluent phase-equivalent pairs and gives closed analytic potentials for fixed-angular-momentum inversion. The explicit 3S1 family (36) with independently adjustable bound-state energy and asymptotic normalization constant is a concrete advance over purely numerical integral implementations. The paper also correctly identifies the 1S0 evaluation problem, which is an honest limitation. The claimed unification is conceptually attractive, but the central step rests on an imported theorem whose hypotheses are not stated or checked, so the significance is contingent on that verification.

major comments (3)

[Sec. 3, Eq. (24) and Eq. (31)] The central formula is obtained by importing the multi-confluent Wronskian theorem of [2] without stating its hypotheses. In the intended application, the M non-confluent transformations produce a potential V_M that is singular at r=0 with singularity parameter n=2 (Eqs. (35) and (38)), and the confluent pair uses a generalized eigenfunction u1 = beta u0_perp + dE u0 that is not the regular solution at the origin and grows at infinity for beta>0. It is not automatic that the theorem of [2] applies to such chains; it may require nonzero Wronskians, node-free seed functions, or specific boundary conditions. The authors should state the theorem explicitly and verify its hypotheses, or give a direct proof of Eq. (31) by induction from Eqs. (18) and (23). Without this, Eq. (31) rests on an unverified import.
[Sec. 4.2, Eq. (39)] The 1S0 application is not demonstrated. The text states that neither symbolic computation, automatic differentiation, nor numerical evaluation could evaluate Eq. (39). Thus the 'whole family of deep potentials' is only a formal expression; no phase-shift or bound-state property is checked. The authors should either provide a stable evaluation scheme (e.g., high-precision arithmetic, an alternative ordering of transformations, or a reduction to a Wronskian with smaller dimension) or verify Eq. (39) in a limiting case where the integral formula (23) is computable, such as M=2 or a degenerate-parameter limit. As it stands, the claimed 1S0 result is unsupported.
[Sec. 3, after Eq. (31)] The sentence 'This formula can be extended to the case of the addition of an arbitrary number of bound states ... simply by adding pairs of functions u0 and u1 in the Wronskian' is asserted without proof or a precise statement of the underlying theorem. Since the abstract promises an 'elegant complete solution' to the inversion problem, this extension is load-bearing. Please state the relevant result of [2] in full, including any conditions on the added energies (distinctness, ordering) and on the Jordan-chain structure, and indicate how multiple bound-state pairs are incorporated. A mere appeal to [2] is insufficient if the hypotheses are not given.

minor comments (4)

[Sec. 2, Eq. (12)] The definition of the Wronskian is misprinted: 'W[ψ,ϕ] ≡ ψϕ′ − ϕ′ψ' is identically zero as written. It should be ψϕ′ − ψ′ϕ (or an equivalent form). The subsequent formulas make the intended definition clear, but the typo should be corrected.
[Sec. 3, Eq. (33)] The identity (33) is stated for an integral from r to infinity, but it requires the factorization solution to decay sufficiently fast at infinity (as in the exponential example). For non-decaying solutions such as sinh(κr) used in Eq. (35), both sides diverge. Please specify the domain of validity of (33).
[Sec. 4.1, Eq. (36)] The condition 'κ2 ≠ κ0 ≠ κ1' is imprecise; it should say the three parameters are pairwise distinct. The limiting cases κ2=κ0 and κ2=κ1 (the latter is treated separately in Eq. (37)) should be discussed or stated to follow by limit.
[Sec. 4.2] The statement that 'automatic differentiation also failed' is vague. It would be helpful to specify the computational environment, the precision used, and the nature of the failure (overflow, singular Wronskians, etc.), or to move this detail to a supplementary material section.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (31) is an application of external confluent-SUSY Wronskian theorems, not a restatement of fitted inputs.

full rationale

The derivation chain is self-contained in the sense that the central formula (31) is obtained by substituting the external generalized-eigenfunction expression (29) from [1] into the external Wronskian theorem (24) from [2]. The preceding integral formula (23) is derived directly from SUSY transformation pairs, and the identity (33) is an independent Wronskian relation, explicitly checkable in the M=0 example and derivable from [17]/[1], rather than a fitted assumption. The parameters κ_i and β are stated inputs of the inverse problem—fixed by phase-shift data and bound-state properties—and the potentials (36), (37), and (39) are outputs, with numerical verification reported in Sec. 4.1. Citations to the authors' prior work ([10,14,15], etc.) provide background, classification tables, and parameter conventions, but they are not load-bearing for Eq. (31), whose validity rests on external theorems [1,2]. The skeptic concern about unstated hypotheses of [2] is a correctness/robustness issue, not circularity: no derivation step is equivalent to its conclusion by construction. The numerical difficulty of evaluating (39) is a technical limitation, not a sign that the result reduces to its inputs.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central formula (31) is assembled from external results: the Crum-Krein Wronskian for non-confluent chains (cited [13], standard), the Wronskian representation of multi-confluent chains (the theorem of [2], not restated), and the generalized-eigenfunction expression u1 = beta u0_perp + dE u0 (from [1]). The phase-shift and spectrum bookkeeping (Tables 1 and 2) is imported from the authors' own prior papers [14, 15], a self-citation cluster that does not by itself weaken the derivation. The free parameters kappa_i and beta/alpha encode the experimental input (phase shifts, deuteron binding energy, ANC): legitimate data for an inverse problem, not hidden fit knobs. The generalization to more bound states is asserted. No new physical entity is introduced: the Jordan-chain pairs (u0, u1) are mathematical devices inherited from [1, 2], and the Moscow-potential idea comes from [21, 22].

free parameters (7)

kappa0 (3S1 factorization parameter) = 0.9090 fm^-1
Fitted to the low-energy 3S1 phase shift; with kappa1 gives scattering length 1/kappa0 + 1/kappa1 = 5.42 fm (Sec. 4.1).
kappa1 (3S1 factorization parameter) = 0.2315 fm^-1
Fitted to the low-energy 3S1 phase shift and effective range; equals the deuteron pseudo wave number at the physical point (Sec. 4.1).
kappa2 (bound-state energy parameter) = free in (36); 0.2315 fm^-1 for the deuteron
Sets the added bound-state energy arbitrarily in the family (36); pinned to 2.225 MeV for the deuteron (Sec. 4.1).
beta (ANC parameter in (36)) = free
Controls the asymptotic normalization constant of the added bound state; must be positive to keep the factorization solution node-free (Sec. 4.1).
alpha (ANC parameter in (37)) = free; alpha = 0 physical
Selects the bound-state ANC in the deuteron case; alpha = 0 gives the shortest-range Eckart potential (Sec. 4.1).
kappa0..kappa5 (1S0 factorization parameters) = 0.6152, 2.0424, 4.1650, 4.6, -0.0401, -0.7540 fm^-1
Taken from the 1S0 phase-shift fit of [19]; reused to build V6 (Eq. 38) and the deep potential (39) (Sec. 4.2).
kappa6, beta (1S0 forbidden bound state) = free
Proposed as tunable parameters to fit further partial waves in a spin/parity-independent search (Sec. 4.2).

assumptions (7)

domain assumption Multi-confluent SUSY transformation chains of arbitrary order are representable by a Wronskian of factorization (generalized eigen) functions, the theorem of [2].
Load-bearing input for Eq. (24) and hence for the main formula (31); cited but its hypotheses are not stated, so applicability to singular np potentials is assumed.
domain assumption The second generalized eigenfunction of the Jordan chain takes the closed form u1 = beta u0_perp + dE u0 (Eq. 29, from [1]).
Turns the abstract Wronskian (24) into the computable form (31). The boundary conditions under which (H - E)u1 = u0 holds are not discussed; the variation-of-parameters argument works for the exponential case used.
domain assumption Transformation effects on phase shift and bound spectrum (Table 1, adapted from [15]): each Tl, Tr, Trem, Tadd changes delta(k) by a known arctangent term.
Guarantees phase equivalence and bound-state addition by construction; consistent with Levinson counting and reproduces the 3S1 scattering length in the application.
standard math Generalized Levinson theorem: delta(0) = N pi and delta(infinity) = -(n - l) pi/2 (cited [16]).
Used in Eqs. (20)-(21) to motivate the singular inverse problem and the replacement of bound states by an increased singularity parameter.
domain assumption Identity (33): integral_r^infinity phi0(t,E)^2 dt = W[phi0(r,E), dE phi0(r,E)], asserted to derive from [17] and [1].
Bridges the integral formulation (23) and the Wronskian formulation (31), supporting the 'most general family' claim; verified in the text only for the exponential case of Sec. 4.1.
domain assumption Starting potential V0 = 0 with S-wave solutions behaving as r^(n0+1), n0 = 0, for both np applications.
Modeling choice (Secs. 4.1-4.2) that makes all factorization solutions elementary (exponentials and hyperbolic sines); the inversion starts from the centrifugal-free case.
ad hoc to paper Equation (31) extends to an arbitrary number of added bound states by adding further (u0, u1) pairs in the Wronskian.
Asserted on the authority of [2] without stating the multi-confluent generalization or its conditions (Sec. 3, after Eq. 31); not demonstrated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics." pith.science (2026). https://pith.science/paper/HILZ2BJW

@misc{pith2026250819022,
  author       = {Pith},
  title        = {Pith review of: Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HILZ2BJW}},
  note         = {Machine review of arXiv:2508.19022}
}

read the original abstract

The Wronskian formulation of supersymmetric quantum mechanics (SUSYQM) confluent transformation pairs is applied to the construction of phase-equivalent potentials with different bound spectra, replacing integral formulas. This allows to unify the two steps of a SUSYQM inversion scheme consisting in (i) the construction of a unique bound-state-less potential, possibly singular, from phase-shift inversion by a chain of non-confluent SUSYQM transformations, and (ii) the phase-equivalent addition of bound states by confluent SUSYQM pairs. Both steps are now combined in a single Wronskian formula, providing an elegant complete solution to the fixedangular-momentum inversion problem. This formalism is applied to the inversion of 3S1 and 1S0 neutron-proton data and its numerical implementation is discussed.

Figures

Figures reproduced from arXiv: 2508.19022 by the authors.

**Figure 1.** Phase-equivalent potentials (36) fitting the low-energy 3S1 proton-neutron phase shifts obtained thanks to the supersymmetric inversion method. The physical potential (37) has α = 0 and κ1 = 0.2315 fm−1 . 4.2. 1S0 state For the singlet state, we fit the phase shift on the whole elastic-scattering range. In [19] (see also [20]), a satisfactory fit was found with 6 non-confluent transformations, starting again from V0… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

[1]

Wronskian diﬀerential formula for conﬂuent supersymmetric quantum mechanics

D Bermudez, DJ Fern´ andez C, and N Fern´ andez-Garc ´ ıa. Wronskian diﬀerential formula for conﬂuent supersymmetric quantum mechanics. Physics Letters A , 376:692–696, 2012

work page 2012
[2]

Wronskian representation for conﬂuent su persymmetric transformation chains of arbitrary order

A Schulze-Halberg. Wronskian representation for conﬂuent su persymmetric transformation chains of arbitrary order. The European Physical Journal Plus , 128:1–17, 2013

work page 2013
[15]

Single- and coupled-channel radial inverse scattering with supersymmetric transformations

D Baye, J-M Sparenberg, AM Pupasov-Maksimov, and BF Samso nov. Single- and coupled-channel radial inverse scattering with supersymmetric transformations. Journal of Physics A: Mathematical and Theoretical , 47:243001, 2014

work page 2014
[3]

Inverse problems in quantum scattering theory

K Chadan and PC Sabatier. Inverse problems in quantum scattering theory . Springer Science & Business Media, 2012

work page 2012
[4]

J. R. Taylor. Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. Wiley, New York, 1972

work page 1972
[5]

Dynamical breaking of supersymmetry

E Witten. Dynamical breaking of supersymmetry. Nuclear Physics B , 188:513–554, 1981

work page 1981
[6]

Supersymmetric methods in quantum and statistical physics

G Junker. Supersymmetric methods in quantum and statistical physics . Springer Science & Business Media, 2012

work page 2012
[7]

Supersymmetry between deep and shallow nucleus-nucle us potentials

D Baye. Supersymmetry between deep and shallow nucleus-nucle us potentials. Physical Review Letters , 58:2738–2741, 1987

work page 1987

Show all 22 references

[8]

Phase-equivalent potentials from supersymmetry

D Baye. Phase-equivalent potentials from supersymmetry. Journal of Physics A: Mathematical and General , 20:5529, 1987

1987
[9]

Most general form of phase-equiv alent radial potentials for arbitrary modiﬁcations of the bound spectrum

D Baye and J-M Sparenberg. Most general form of phase-equiv alent radial potentials for arbitrary modiﬁcations of the bound spectrum. Physical Review Letters, 73:2789, 1994

1994
[10]

Sparenberg and D

J.-M. Sparenberg and D. Baye. Inverse scattering with singula r potentials: A supersymmetric approach. Phys. Rev. C , 55:2175–2184, 1997

1997
[11]

Supersymmetric quantum mechanics of one-dimen sional systems

CV Sukumar. Supersymmetric quantum mechanics of one-dimen sional systems. Journal of Physics A: Mathematical and General , 18:2917, 1985

1985
[12]

Supersymmetric quantum mechanics and the inver se scattering method

CV Sukumar. Supersymmetric quantum mechanics and the inver se scattering method. Journal of Physics A: Mathematical and General , 18:2937, 1985

1985
[13]

Associated Sturm-Liouville systems

MM Crum. Associated Sturm-Liouville systems. The Quarterly Journal of Mathematics, 6:121–127, 1955

1955
[14]

Inverse scattering with supersy mmetric quantum mechanics

D Baye and J-M Sparenberg. Inverse scattering with supersy mmetric quantum mechanics. Journal of Physics A: Mathematical and General , 37:10223, 2004. REFERENCES 13

2004
[16]

Asymptotic phase-shifts and bound states for two-b ody central interactions

P Swan. Asymptotic phase-shifts and bound states for two-b ody central interactions. Nuclear Physics , 46:669–694, 1963

1963
[17]

Hyperconﬂuent third-order supersymmetric quantum mechanics

Encarnaci´ on Salinas-Hern´ andez et al. Hyperconﬂuent third-order supersymmetric quantum mechanics. Journal of Physics A: Mathematical and Theoretical , 44(36):365302, 2011

2011
[18]

The low-energy n eutron-proton scattering parameters and the deuteron

JJ de Swart, CPF Terheggen, and VGJ Stoks. The low-energy n eutron-proton scattering parameters and the deuteron. arXiv preprint nucl-th/9509032 , 1995

1995 arXiv
[19]

Phase equivalent chains of Darboux transformations in scattering theory

BF Samsonov and F Stancu. Phase equivalent chains of Darboux transformations in scattering theory. Physical Review C , 66:034001, 2002

2002
[20]

Supersymmetric inversion of eﬀective-range expansions

B Midya, J Evrard, S Abramowicz, OL Ram ´ ırez Su´ arez, and J-M Sparenberg. Supersymmetric inversion of eﬀective-range expansions. Physical Review C , 91:054004, 2015

2015
[21]

Symmetry in the nucleon-nucleo n interaction and the model of Moscow potential

VI Kukulin and VN Pomerantsev. Symmetry in the nucleon-nucleo n interaction and the model of Moscow potential. Progress in Theoretical Physics , 88:159–209, 1992

1992
[22]

Toward a spin-and parity-independent nucle on-nucleon potential

J-M Sparenberg. Toward a spin-and parity-independent nucle on-nucleon potential. Europhysics Letters, 59:507, 2002

2002

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Wronskian diﬀerential formula for conﬂuent supersymmetric quantum mechanics

D Bermudez, DJ Fern´ andez C, and N Fern´ andez-Garc ´ ıa. Wronskian diﬀerential formula for conﬂuent supersymmetric quantum mechanics. Physics Letters A , 376:692–696, 2012

work page 2012

[2] [2]

Wronskian representation for conﬂuent su persymmetric transformation chains of arbitrary order

A Schulze-Halberg. Wronskian representation for conﬂuent su persymmetric transformation chains of arbitrary order. The European Physical Journal Plus , 128:1–17, 2013

work page 2013

[3] [15]

Single- and coupled-channel radial inverse scattering with supersymmetric transformations

D Baye, J-M Sparenberg, AM Pupasov-Maksimov, and BF Samso nov. Single- and coupled-channel radial inverse scattering with supersymmetric transformations. Journal of Physics A: Mathematical and Theoretical , 47:243001, 2014

work page 2014

[4] [3]

Inverse problems in quantum scattering theory

K Chadan and PC Sabatier. Inverse problems in quantum scattering theory . Springer Science & Business Media, 2012

work page 2012

[5] [4]

J. R. Taylor. Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. Wiley, New York, 1972

work page 1972

[6] [5]

Dynamical breaking of supersymmetry

E Witten. Dynamical breaking of supersymmetry. Nuclear Physics B , 188:513–554, 1981

work page 1981

[7] [6]

Supersymmetric methods in quantum and statistical physics

G Junker. Supersymmetric methods in quantum and statistical physics . Springer Science & Business Media, 2012

work page 2012

[8] [7]

Supersymmetry between deep and shallow nucleus-nucle us potentials

D Baye. Supersymmetry between deep and shallow nucleus-nucle us potentials. Physical Review Letters , 58:2738–2741, 1987

work page 1987