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

Bachkhaznadji, A.; Lassaut, M.

doi:10.1007/s00601-013-0696-z

Cited by 9 publications

(14 citation statements)

References 30 publications

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“…In this subsection a solvable many particle model with four-body interaction in one dimension in the presence of balanced loss and gain terms is investigated. Many-body systems with fourbody interaction have been considered earlier in the literature [45,46,47]. The exactly solvable quantum models of Calogero and Sutherland type with translationally invariant two and fourbody interactions is investigated in Ref.…”

Section: Calogero-type Systems With Four-body Interactionmentioning

confidence: 99%

“…An exactly solvable four-body interaction with non-translationally invariant interactions is discussed in Ref. [47]. In our example, the four-body inverse square interaction is generated in the presence of balanced loss and gain terms in the original coordinate x i by considering a Calogero-type of potential for the reduced system in z i coordinates.…”

Section: Calogero-type Systems With Four-body Interactionmentioning

confidence: 99%

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Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

Ghosh

Sinha

2018

Annals of Physics

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A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

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Section: Calogero-type Systems With Four-body Interactionmentioning

confidence: 99%

Section: Calogero-type Systems With Four-body Interactionmentioning

confidence: 99%

Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

Ghosh

Sinha

2018

Annals of Physics

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“…It should be mentioned that the Stoke wedge for which the wave function is normalizable is not unique. Any possible solution satisfying condition (32) gives a Stoke wedge in which the wave function is normalizable. For example, for m = 2, one needs to have cos(2θ 1 ) + cos(2θ 2 ) < 0 in order to get a normalizable solution.…”

Section: Simple Harmonic Oscillatormentioning

confidence: 99%

“…[31] and exactly solvable four-body interaction with translational non-invariant interactions is discussed in Ref. [32]. The form of the four-body interactions in these models are different from those presented in Ref.…”

Section: Introductionmentioning

confidence: 99%

On the bound states and correlation functions of a class of Calogero-type quantum many-body problems with balanced loss and gain

Sinha¹,

Ghosh²

2019

J. Phys. A: Math. Theor.

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The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translational invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.

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“…those where the potential is of the form "oscillator/inverse square") have received considerable attention, and many interesting properties have been discovered [53][54][55][56][57][58][59][60][61][62][63][64][65]. There are also many works which have attempted to obtain new ES models by extending existing ones through separation of variables [66][67][68]. More complicated extensions, which have connections with orthogonal polynomials, can be obtained by PT (parity and time reversal) symmetric quantum mechanics [69][70][71].…”

Section: Introductionmentioning

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

Cited by 9 publications

References 30 publications

Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

On the bound states and correlation functions of a class of Calogero-type quantum many-body problems with balanced loss and gain

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

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