Dressing chains and the spectral theory of the Schr�dinger operator

Веселов, А. П.; Shabat, A. B.

doi:10.1007/bf01085979

Cited by 289 publications

(432 citation statements)

References 15 publications

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“…We would like to point out what in our opinion is new in respect to [16]. First, we introduced in the dressing chain (in their language) irreducible transformations which correspond to quantum mechanical systems with only one sequence of levels.…”

Section: Discussionmentioning

confidence: 99%

“…This approach reflects in part results obtained in [16] in their construction of dressing chains in connection with integrable bi-hamiltonian systems and in [17] in the context of numerical illustrations of (cyclic) shape-invariance preserving standard shape invariance in each step. However our approach also leads to new findings concerning explicitly with the role of irreducible second order Darboux transformations [7].…”

Section: Introductionmentioning

confidence: 99%

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Systems with higher-order shape invariance: spectral and algebraic properties

et al. 2000

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We study a complex intertwining relation of second order for Schrödinger operators and construct third order symmetry operators for them. A modification of this approach leads to a higher order shape invariance. We analyze with particular attention irreducible second order Darboux transformations which together with the first order act as building blocks. For the third order shape-invariance irreducible Darboux transformations entail only one sequence of equidistant levels while for the reducible case the structure consists of up to three infinite sequences of equidistant levels and, in some cases, singlets or doublets of isolated levels.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Systems with higher-order shape invariance: spectral and algebraic properties

et al. 2000

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“…Painlevé IV equation was first represented as a simple system of three first order equations (dressing chain) in [28]. Such a symmetric form of PIV was used in [22] to obtain all the rational solutions of the equation in the remarkable determinant form (simultaneously with the independent work [16]).…”

Section: Explicit Solutions For Even Multiplicitymentioning

confidence: 99%

Painlevé IV and degenerate Gaussian unitary ensembles

Chen

Feigin

2006

J. Phys. A: Math. Gen.

123

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We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t , this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable. † ychen@ic.ac.uk

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“…For q = −1 one gets similar situations with the functions f j (x) obeying certain parity symmetry [47]. If N −1 m=0 µ m = 0 then for odd N and some cases of even N the potential u 0 (x) is expressed through the hyperelliptic functions [50,51].…”

Section: Self-similar Potentialsmentioning

confidence: 96%

“…Then for N = 1, 2 one easily gets the potentials u 0 (x) ∝ x 2 , ax 2 + b/x 2 . For N = 3 the corresponding system of equations for f j (x) provides a "cyclic" representation of the Painlevé-IV equation [8,50].…”

Section: Self-similar Potentialsmentioning

confidence: 99%

The Factorization Method, Self-Similar Potentials and Quantum Algebras

Spiridonov¹

2001

Special Functions 2000: Current Perspective and Future Directions

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The factorization method is a convenient operator language formalism for consideration of certain spectral problems. In the simplest differential operators realization it corresponds to the Darboux transformations technique for linear ODE of the second order. In this particular case the method was developed by Schrödinger and became well known to physicists due to the connections with quantum mechanics and supersymmetry. In the theory of orthogonal polynomials its origins go back to the Christoffel's theory of kernel polynomials, etc. Special functions are defined in this formalism as the functions associated with similarity reductions of the factorization chains.We consider in this lecture in detail the Schrödinger equation case and review some recent developments in this field. In particular, a class of self-similar potentials is described whose discrete spectrum consists of a finite number of geometric progressions. Such spectra are generated by particular polynomial quantum algebras which include q-analogues of the harmonic oscillator and su(1, 1) algebras. Coherent states of these potentials are described by differential-delay equations of the pantograph type. Applications to infinite soliton systems, Ising chains, random matrices, and lattice Coulomb gases are briefly outlined.1 2000 Mathematical Subject Classification: primary 33D80, 39A10; secondary 81R30, 82B20.

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Dressing chains and the spectral theory of the Schr�dinger operator

Cited by 289 publications

References 15 publications

Systems with higher-order shape invariance: spectral and algebraic properties

Systems with higher-order shape invariance: spectral and algebraic properties

Painlevé IV and degenerate Gaussian unitary ensembles

The Factorization Method, Self-Similar Potentials and Quantum Algebras

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