The phenomenon of Darboux displacements

Fernández, J C David; Mielnik, Bogdan; Rosas-Ortiz, Óscar; Samsonov, B. F.

doi:10.1016/s0375-9601(01)00839-8

Cited by 41 publications

(60 citation statements)

References 30 publications

(104 reference statements)

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“…As both families of quantum systems are characterized by nontrivial, higher derivative integrals of motion, one could expect that supersymmetric extensions of them should possess some peculiar properties. This is indeed the case [16,17,18,19,20,21], and exotic supersymmetric structures of reflectionless and finite-gap systems found recently some interesting physical applications [22,23,24,25,26].…”

Section: Introductionmentioning

confidence: 86%

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

2013

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We investigate a peculiar supersymmetry of the pairs of reflectionless quantum mechanical systems described by n-soliton potentials of a general form that depends on n scaling and n translation parameters. We show that if all the discrete energy levels of the subsystems are different, the superalgebra, being insensitive to translation parameters, is generated by two supercharges of differential order 2n, two supercharges of order 2n + 1, and two bosonic integrals of order 2n+1 composed from Lax integrals of the partners. The exotic supersymmetry undergoes a reduction when r discrete energy levels of one subsystem coincide with any r discrete levels of the partner, the total order of the two independent intertwining generators reduces then to 4n − 2r + 1, and the nonlinear superalgebraic structure acquires a dependence on r relative translations. For a complete pairwise coincidence of the scaling parameters which control the energies of the bound states and the transmission scattering amplitudes, the emerging isospectrality is detected by a transmutation of one of the Lax integrals into a bosonic central charge. Within the isospectral class, we reveal a special case giving a new family of finite-gap first order Bogoliubov-de Gennes systems related to the AKNS integrable hierarchy.

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Section: Introductionmentioning

confidence: 86%

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

2013

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“…We might add here that applying second order Darboux transformations to the Lamé potentials, Fernández et al [15] have obtained an interesting nonlocal effect, by which the transformed potential becomes an exact or approximately displaced copy of the original one.…”

Section: Lamé and Associated Lamé Potentialsmentioning

confidence: 96%

“…On the other hand, for a = 1 one can show [8] that the three potentials are not distinct but are self-isospectral (i.e. differ from each other by a translation by a constant) [13,15].…”

Section: Pt Invariant Potentials With Real Band Spectramentioning

confidence: 99%

Periodic potentials and supersymmetry

Khare¹,

Sukhatme²

2004

J. Phys. A: Math. Gen.

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We review the current status of one dimensional periodic potentials and also present several new results. It is shown that using the formalism of supersymmetric quantum mechanics, one can considerably enlarge the limited class of analytically solvable one-dimensional periodic potentials.Further, using the Landen transformations as well as cyclic identities for Jacobi elliptic functions discovered by us recently, it is shown that a linear superposition of Lamé (as well as associated Lamé) potentials are also analytically solvable. Finally, using anti-isospectral transformations, we also obtain a class of analytically solvable, complex, PT-invariant, periodic potentials having real band spectra.

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“…Notice that there are three regions in which (19) yields a non-singular Darboux transformation (4), Ω 1 = {γ 3 , γ 4 < γ 1 }, Ω 2 = {γ 1 < γ 3 , γ 4 < γ 2 }, Ω 3 = {γ 3 , γ 4 > γ 2 }, but only in Ω 2 can one achieve the consistency between (21) and (22), visualized by the intersection of two surfaces represented on Fig. 2.…”

Section: The Translational Effects Of the Darboux Operationsmentioning

confidence: 99%

“…In its most inconspicuous form, it produces an interesting non-local effect: the transformed potential becomes an exact or approximate displaced copy of the initial one, a phenomenon which we call the translational invariance with respect to Darboux transformations or shortly Darboux invariance [21]. Quite significantly, the effect shows itself asymptotically even if the periodicity of the initial potential is not preserved.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal supersymmetric deformations of periodic potentials

Mielnik

Rosas-Ortiz

et al. 2002

J. Phys. A: Math. Gen.

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Abstract. Irreducible 2-order Darboux transformations are applied to the periodic Schrödinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials with periodicity defects and bound states embedded into the spectral gaps. The method is applied to the Lamé and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.

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The phenomenon of Darboux displacements

Cited by 41 publications

References 30 publications

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

Periodic potentials and supersymmetry

Nonlocal supersymmetric deformations of periodic potentials

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