Chains of Darboux transformations for the matrix Schrödinger equation

Samsonov, B. F.; Pecheritsin, A.A.

doi:10.1088/0305-4470/37/1/016

Cited by 30 publications

(88 citation statements)

References 13 publications

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“…The same options are available for the matrix Darboux transformations, though a systematic study of all possible chains is not done yet for the coupled-channel case. For the case of equal thresholds and different factorization energies, an extension of the Crum-Krein formulas to the matrix case was developed in [87]. For the purpose of inversion problems, it is often necessary to combine one-channel transformations which are able to invert a single phase shift with matrix transformations which are able to invert a full scattering matrix.…”

Section: Chains Of Transformationsmentioning

confidence: 99%

“…Such single-channel transformations cannot be directly incorporated into the usual chain of matrix supersymmetric transformations considered in [87]. Nevertheless, theorems proved in [87] have a more general validity than supersymmetric transformations of the matrix Schrödinger equation. Actually, they represent a closure of a special recursion procedure which can easily be generalized to incorporate both single-channel and matrix transformations.…”

Section: Chains Of Transformationsmentioning

confidence: 99%

“…Actually, they represent a closure of a special recursion procedure which can easily be generalized to incorporate both single-channel and matrix transformations. This generalization of the coupled-channel Crum-Krein formulas established in [87] requires to formulate single-channel transformations as singular matrix transformations [88].…”

Section: Chains Of Transformationsmentioning

confidence: 99%

“…To incorporate singular transformations of the type (5.79) into the chain of usual matrix supersymmetric transformations considered in [87], we need auxiliary transformation matrices U which are built from single-channel transformation functions…”

Section: Chains Of Transformationsmentioning

confidence: 99%

See 3 more Smart Citations

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

Baye¹,

Sparenberg²,

Pupasov-Maksimov³

et al. 2014

J. Phys. A: Math. Theor.

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Abstract. The present status of the three-dimensional inverse-scattering method with supersymmetric transformations is reviewed for the coupled-channel case. We first revisit in a pedagogical way the single-channel case, where the supersymmetric approach is shown to provide a complete, efficient and elegant solution to the inverse-scattering problem for the radial Schrödinger equation with short-range interactions. A special emphasis is put on the differences between conservative and non-conservative transformations, i.e. transformations that do or do not conserve the behaviour of solutions of the radial Schrödinger equation at the origin. In particular, we show that for the zero initial potential, a non-conservative transformation is always equivalent to a pair of conservative transformations. These single-channel results are illustrated on the inversion of the neutron-proton triplet eigenphase shifts for the S and D waves.We then summarize and extend our previous works on the coupled-channel case, i.e. on systems of coupled radial Schrödinger equations, and stress remaining difficulties and open questions of this problem by putting it in perspective with the single-channel case. We mostly concentrate on two-channel examples to illustrate general principles while keeping mathematics as simple as possible. In particular, we discuss the important difference between the equal-threshold and different-threshold problems. For equal thresholds, conservative transformations can provide non-diagonal Jost and scattering matrices. Iterations of such transformations in the two-channel case are studied and shown to lead to practical algorithms for inversion. A convenient particular technique where the mixing parameter can be fitted without modifying the eigenphases is developed with iterations of pairs of conjugate transformations. This technique is applied to the neutron-proton triplet S-D scattering matrix, for which exactly-solvable matrix potential models are constructed. For different thresholds, conservative transformations do not seem to be able to provide a non-trivial coupling between channels. In contrast, a single non-conservative transformation can generate coupled-channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled-channel inverse problem with threshold differences.

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Section: Chains Of Transformationsmentioning

confidence: 99%

Section: Chains Of Transformationsmentioning

confidence: 99%

Section: Chains Of Transformationsmentioning

confidence: 99%

Section: Chains Of Transformationsmentioning

confidence: 99%

See 2 more Smart Citations

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

Baye¹,

Sparenberg²,

Pupasov-Maksimov³

et al. 2014

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In general, when a Darboux transformation is applied to a scalar potential, the resulting potential is not scalar, but below we formulate additional conditions that will prevent this (see [8,10] for details).…”

Section: Darboux Transformation For a Scalar Potentialmentioning

confidence: 99%

New exactly solvable periodic potentials for the Dirac equation

et al. 2003

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Abstract.A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig-Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.

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Polynomial pseudosupersymmetry underlying a two-level atom in an external electromagnetic field

2005

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Chains of Darboux transformations for the matrix Schrödinger equation

Abstract: Chains of Darboux transformations for the matrix Schrödinger equation are considered. Matrix generalization of the well-known for the scalar equation Crum-Krein formulas for the resulting action of such chains is given.

Cited by 30 publications

References 13 publications

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

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

New exactly solvable periodic potentials for the Dirac equation

Polynomial pseudosupersymmetry underlying a two-level atom in an external electromagnetic field

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