A model of the Calogero type in the<i>D</i>-dimensional space

Bachkhaznadji, A.; Lassaut, M.; Lombard, R.J.

doi:10.1088/1751-8113/40/30/012

Cited by 8 publications

(6 citation statements)

References 14 publications

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“…with n = 0, 1, 2, ... a = √ 1 + 4λ 2 (15) The extension to the whole interval [0, 2π] is made in two steps. First one, from [0, π] up to [π, 2π] using the periodicity of the solutions.…”

Section: A Generalization Of the Linear Wolfes Four-body Problemmentioning

confidence: 99%

“…A new integrable model on the line of the Calogero type with a non-translationally invariant two-body potential, was worked out by Diaf et al [14]. The extension of this linear model to D-dimensional space was done in [15]. We note also that a generalization of this latter linear model was solved by Meljanac et al [16], by emphasizing the underlying conformal SU (1, 1) symmetry of the model.…”

Section: Introductionmentioning

confidence: 98%

“…For positive values a, c, d, the quantity β + D ℓ,m,n is minimal for n = 0, m = 0, ℓ = 0 and a = c = d = 0 ( recall that a ≥ 0, see (15), that c ≥ 0, see (20) and that d ≥ 0 see (25)). It puts constraint on β to satisfy β > −9 when a,c and d are positive.…”

Section: Introducing the Parameter Bmentioning

confidence: 99%

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Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

Bachkhaznadji

Lassaut

2013

Few-Body Syst

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We propose and solve exactly the Schrödinger equation of a bound quantum system consisting in four particles moving on a real line with both translationally invariant four particles interactions of Wolfes type [1] and additional non translationally invariant four-body potentials. We also generalize and solve exactly this problem in any D-dimensional space by providing full eigensolutions and the corresponding energy spectrum. We discuss the domain of the coupling constant where the irregular solutions becomes physically acceptable PACS: 02.30.Hq, 03.65.-w, 03.65.GeThe paper is organized as follows. In section 2 we expose and solve the problem for the linear case. The section 3 is devoted to extension to D-dimensional problem. Our conclusions are drawn in section 4.

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Section: A Generalization Of the Linear Wolfes Four-body Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

Bachkhaznadji

Lassaut

2013

Few-Body Syst

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“…The early works of Calogero and Sutherland have been extended to many more complex systems, concerning two-and three-body problems. We can quote, in a non exhaustive way, the works of several authors [10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable $N$-body quantum systems with $N=3^k \ ( k \geq 2)$ in the $D=1$ dimensional space

Bachkhaznadji,

Lassaut

2016

Preprint

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We study the exact solutions of a particular class of N confined particles of equal mass, with N = 3 k (k = 2, 3, ...), in the D = 1 dimensional space. The particles are clustered in clusters of 3 particles. The interactions involve a confining mean field, two-body Calogero type of potentials inside the cluster, interactions between the centres of mass of the clusters and finally a non-translationally invariant N-body potential. The case of 9 particles is exactly solved, in a first step, by providing the full eigensolutions and eigenenergies. Extending this procedure, the general case of N particles (N = 3 k , k ≥ 2) is studied in a second step. The exact solutions are obtained via appropriate coordinate transformations and separation of variables. The eigenwave functions and the corresponding energy spectrum are provided.

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“…This 9-body model is analogue to the one in the forth paper of [82], up to a mean field term. In fact, one can use this algorithm and add a mean field to recover the 3 k -body model mentioned in [68,73,74,82].…”

Section: Now Introducing 3 Variables Qmentioning

confidence: 99%

New families of exactly solvable many-body models and superintegrable systems

Chen¹

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In this thesis, we study various extensions of Calogero models and superintegrable systems. We construct a kN-body one-dimensional model which reduces to the familiar Calogero model when k = 1. We present a class of many-body systems that are equivalent to harmonic oscillators. We study interesting extensions of the D-dimensional Coulomb-Kepler system and show that when the extension satisfies certain conditions, then some components of the Laplace-Runge-Lenz vector can be extended to conserved quantity of the new models. By introducing block separation of variables, we construct the Kepler-singular oscillator type models which are a new family of superintegrable systems. We also use separation of variables to obtain the energy spectrum, eigenfunctions and corresponding quadratic algebraic structures. Separation of variables is not only used for solving eigenvalue problems but also provides us with new tools for generalizing superintegrable systems. We generalize the double harmonic-singular oscillators and Kepler-singular oscillator systems. The integrals of new models now can involve angles from block spherical coordinates. A derivation of more general quadratic algebraic structures is presented as well. We also give examples to show they can be solved in terms of so-called X 1 Jacobi polynomials. ABSTRACT iii Declaration by authorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

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A model of the Calogero type in theD-dimensional space

Cited by 8 publications

References 14 publications

Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

Exactly solvable $N$-body quantum systems with $N=3^k \ ( k \geq 2)$ in the $D=1$ dimensional space

New families of exactly solvable many-body models and superintegrable systems

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