New classes of Schrödinger equations equivalent to the free particle equation through non-local transformations

Bluman, George W.; Shtelen, V. M.

doi:10.1088/0305-4470/29/15/018

Cited by 21 publications

(24 citation statements)

References 5 publications

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“…The difference between the free case and the oscillator case lies in the fact that the time-derivative operator is, in the first case, associated with a positive root of the sl(2) subalgebra, as in formula (5). In the second case it is associated with the Cartan generator (for the linear potential, the time-derivative operator is a symmetry generator which does not coincide with a generator of the sl(2) subalgebra [37,1]).…”

Section: The Invariant Pde Of Degreementioning

confidence: 99%

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

Aizawa

Kuznetsova

Toppan³

2015

Journal of Mathematical Physics

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We construct, for any given ℓ = 1 2 + N 0 , the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra.At the given ℓ, two invariant equations in one time and ℓ + 2 ). The spectrum of the ℓ-oscillator, derived from a specific osp(1|2ℓ + 1) h.w.r., is explicitly presented.The two sets of invariant PDEs are determined by imposing (representationdependent) on-shell invariant conditions both for degree 1 operators (those with continuum spectrum) and for degree 0 operators (those with discrete spectrum).The on-shell condition is better understood by enlarging the Conformal Galilei Algebras with the addition of certain second-order differential operators. Two compatible structures (the algebra/superalgebra duality) are defined for the enlarged set of operators. *

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Section: The Invariant Pde Of Degreementioning

confidence: 99%

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

Aizawa

Kuznetsova

Toppan³

2015

Journal of Mathematical Physics

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“…respectively. Now, in a second step we make a non-local transformation [17] of the independent variables (x, t) → (y, τ ) with…”

Section: Reducible and Irreducible Second Order Intertwiningmentioning

confidence: 99%

Intertwining relations of non-stationary Schrödinger operators

Cannata¹,

Ioffe²,

Junker³

et al. 1999

J. Phys. A: Math. Gen.

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General first-and higher-order intertwining relations between non-stationary one-dimensional Schrödinger operators are introduced. For the first-order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in a R-separation of variables. The Fokker-Planck and diffusion equation are briefly considered. Second-order intertwining operators are also discussed within a general approach. However, due to its complicated structure only particular solutions are given in some

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“…Corollary. Let e −χ = e −χ 0 −iχ 1 be a solution of the TDSE with potential V 0 (x, t), with χ satisfying the reality condition (4). Let D and T denote, respectively, the Darboux transformation (3) and the point transformation (8) defined by χ via eqns.…”

Section: The Reality Condition and The Form-preserving Point Tramentioning

confidence: 99%

On form-preserving transformations for the time-dependent Schrödinger equation

Finkel

González‐López

Kamran

et al. 1999

Journal of Mathematical Physics

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In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the timedependent Schrödinger equation ͑TDSE͒. In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of timeindependent potentials related by a usual Darboux transformation for the stationary Schrödinger equation. Thus, any ͑real͒ potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving point transformation of the TDSE to a time-independent potential. The pre-eminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials.

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New classes of Schrödinger equations equivalent to the free particle equation through non-local transformations

Cited by 21 publications

References 5 publications

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

Intertwining relations of non-stationary Schrödinger operators

On form-preserving transformations for the time-dependent Schrödinger equation

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