Supersymmetric quantum mechanics and the inverse scattering method

Sukumar, C V

doi:10.1088/0305-4470/18/15/021

Cited by 173 publications

(148 citation statements)

References 7 publications

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“…In [42] we have shown that the use of phase equivalent potentials [43][44][45][46] is an appropriate method to implement the Pauli principle in three-body calculations based on the adiabatic hyperspherical approach.…”

Section: The Pauli Principle and Complex Scaled Phase Equivalent Potementioning

confidence: 99%

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Dipole excited states in 11Li with complex scaling

Garrido¹,

Fedorov²,

Jensen³

2002

Nuclear Physics A

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The 1 − excitations of the three-body halo nucleus 11 Li are investigated. We use adiabatic hyperspherical expansion and solve the Faddeev equations in coordinate space. The method of complex scaling is used to compute the resonance states. The Pauli forbidden states occupied by core neutrons are excluded by constructing corresponding complex scaled phase equivalent two-body potentials. We use a recently derived neutron-core interaction consistent with known structure and reaction properties of 10 Li and 11 Li. The computed dipole excited states with J π = 1/2 + , J π = 3/2 + , and J π = 5/2 + have energies ranging from 0.6 MeV to 1.0 MeV and widths between 0.15 MeV and 0.65 MeV. We investigate the dependence of the complex energies of these states on the 10 Li spectrum. The finite spin 3/2 of the core and the resulting core-neutron spin-spin interaction are important. The connection with Coulomb dissociation experiments is discussed and a need for better measurements is pointed out. : 21.45.+v, 25.10.+s, 25.60.Gc PACS

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Section: The Pauli Principle and Complex Scaled Phase Equivalent Potementioning

confidence: 99%

“…Coulomb dissociation of 11 Li on Pb at 28, 43 and 280 MeV/nucleon shows a prominent peak at about 1 MeV in the differential B(E1)-distribution in several experiments [9][10][11]. Unfortunately the results of these experiments differ significantly from each other.…”

Section: Introductionmentioning

confidence: 99%

Dipole excited states in 11Li with complex scaling

Garrido¹,

Fedorov²,

Jensen³

2002

Nuclear Physics A

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“…This procedure applies to both the singleand coupled-channel cases [2,3]. In the single-channel case, the same potentials can be constructed with the help of supersymmetric quantum mechanics (SUSYQM) [4,5]. This method is much simpler than the integral-equation one, as the potential is directly related to its scattering-matrix poles.…”

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confidence: 99%

Reconstructing the Nucleon-Nucleon Potential by a New Coupled-Channel Inversion Method

et al. 2011

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A second-order supersymmetric transformation is presented, for the two-channel Schrödinger equation with equal thresholds. It adds a Breit-Wigner term to the mixing parameter, without modifying the eigenphase shifts, and modifies the potential matrix analytically. The iteration of a few such transformations allows a precise fit of realistic mixing parameters in terms of a Padé expansion of both the scattering matrix and the effective-range function. The method is applied to build an exactly-solvable potential for the neutron-proton 3 S1-3 D1 case.PACS numbers: 03.65. Nk, 03.65.Ge, 13.75.Cs In quantum scattering theory, the fundamental inverse problem consists in deducing the interaction potential between two colliding particles from their experimental elastic-scattering cross sections [1]. These cross sections have first to be parametrized in terms of energy-dependent partial-wave phase shifts or scattering matrices. For a central interaction V (r), the partial waves decouple and a sequence of single-channel inverse problems have to be solved. For more complicated interactions, like the tensor interaction in nuclear physics, partial waves may be coupled and matrix potentials have to be constructed from coupled-channel scattering matrices.A formal solution to these inverse problems can be written in terms of integral equations [1]. In practical applications, experimental data turn out to be precisely parametrizable in terms of separable kernels for these equations. These correspond to scattering matrices that are rational functions of the wave number. The integral equations can then be solved analytically and the corresponding potentials are also expressed in a separable form. This procedure applies to both the singleand coupled-channel cases [2,3]. In the single-channel case, the same potentials can be constructed with the help of supersymmetric quantum mechanics (SUSYQM) [4,5]. This method is much simpler than the integral-equation one, as the potential is directly related to its scattering-matrix poles. Moreover, a small number of poles is in general sufficient to fit experimental data on the whole elastic-scattering energy range; this is in particular the case for the neutron-proton singlet (spin 0) channels [6,7], that decouple because of the vanishing tensor interaction.The present Letter aims at extending this very efficient single-channel method to the two-channel case without threshold difference, and at applying it to the neutron-proton triplet (spin 1) coupled channels. This system was studied in the framework of the integral-equation method by Newton and Fulton [2], at low energy, and by Kohlhoff and von Geramb [3], on the whole elastic-scattering range. The present paper subsumes these works, by separating the effect of the coupling between channels in the inversion procedure, by parametrizing the data with a minimal number of poles, and by deriving simple expressions for the corresponding matrix potential.We consider two channels with equal thresholds and angular momenta l and l + 2. The scatter...

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“…The fact that such families exist has been known for a long time from the inverse scattering approach [67], but the Gelfand-Levitan approach to finding them is technically much more complicated than the supersymmetry approach described here. Indeed, the advent of SUSY QM has produced a revival of interest in the determination of isospectral potentials [65,68,69,156,157,158] [66,70,72,75]. In Sec.…”

Section: Isospectral Hamiltoniansmentioning

confidence: 99%

“…In yet another development, several people showed that SUSY QM offers a simple way of obtaining isospectral potentials by using either the Darboux [64] or Abraham-Moses [65] or Pursey [66] techniques, thereby offering glimpses of the deep connection between the methods of the inverse quantum scattering [67], and SUSY QM [68,69,70,71,72]. The intimate connection between the soliton solutions of the KdV hierarchy and SUSY QM was also brought out at this time [73,74,75,76].…”

Section: Introductionmentioning

confidence: 99%

Parasupersymmetry in quantum mechanics

Khare

1993

Journal of Mathematical Physics

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In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all 1 have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

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Supersymmetric quantum mechanics and the inverse scattering method

Cited by 173 publications

References 7 publications

Dipole excited states in 11Li with complex scaling

Dipole excited states in 11Li with complex scaling

Reconstructing the Nucleon-Nucleon Potential by a New Coupled-Channel Inversion Method

Parasupersymmetry in quantum mechanics

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