Newton-Equivalent Hamiltonians for the Harmonic Oscillator

Degasperis, A.; Ruijsenaars, S. N. M.

doi:10.1006/aphy.2001.6169

Cited by 36 publications

(37 citation statements)

References 26 publications

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“…It was only much later -after the theory of Lie superalgebra's was completed -that Palev [12] observed that classes of WQS-solutions for the n-dimensional harmonic oscillator are described by means of representations of the Lie superalgebras osp(1|2n) and sl(1|n) or gl(1|n). This algebraic or representation theoretic approach to quantum systems has revived the interest in WQS's [13,14,15]. The WQS approach has so far been applied to simple systems of free harmonic oscillators, with some interesting and surprising results [16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

Lievens

Stoilova

Jeugt

2006

Journal of Mathematical Physics

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We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M ). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M ). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain. †

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Section: Introductionmentioning

confidence: 99%

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

Lievens

Stoilova

Jeugt

2006

Journal of Mathematical Physics

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“…In this section we discuss two examples, the Meixner-Pollaczek polynomials and the continuous Hahn polynomials, in their full generality. In our previous work on discrete quantum mechanics [14,15], only the special case of the Meixner-Pollaczek polynomials with the phase angle φ = π/2 and the special case of the continuous Hahn polynomials with two real parameters a 1 and a 2 are discussed, partly because these special cases of the two polynomials appear in several other dynamical contexts [6,3,2] and, in particular, they appear in the description of the equilibrium positions [19,13,14] of the classical Ruijsenaars-Schneider van Diejen systems [20,27].…”

Section: Hamiltonian Formulation For Dynamics Of Hypergeometric Orthomentioning

confidence: 99%

“…For a ≤ 0, however, there appear other singularities of φ 0 (x; a), which break the hermiticity. The special case discussed in [14,6,2,3] is β = 0 or φ = π/2. The groundstate wavefunction φ 0 , as annihilated by the operator A, Aφ 0 = 0, is given by…”

Section: Meixner-pollaczek Polynomialsmentioning

confidence: 99%

New Quasi-Exactly Solvable Difference Equation

Sasaki¹

2008

JNMP

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Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the Schrödinger and Heisenberg picture levels. A new quasiexactly solvable difference equation is constructed by crossing these two dynamics, that is, the quadratic potential function of the continuous Hahn polynomials is multiplied by the constant phase factor of the Meixner-Pollaczek type. Its ordinary quantum mechanical counterpart, if exists, does not seem to be known.

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“…Therefore, we obtain the same Newton equation as for the nonrelativistic Hamiltonian (also see [15], [16] in this connection)…”

Section: The a 1 Case Revisitedmentioning

confidence: 99%

The classical hyperbolic Askey-Wilson dynamics without bound states

Ruijsenaars

2008

Theor Math Phys

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We construct an explicit action-angle map for the classical hyperbolic Askey-Wilson dynamics (also known as the relativistic BC1 Calogero-Moser system), restricting the four coupling constants such that no bound states occur. Its features are exact classical analogues of features previously obtained for the quantum eigenfunction transformation. Our construction involves neither scattering theory nor a Lax matrix; instead, it is based on a certain self-duality identity.

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Newton-Equivalent Hamiltonians for the Harmonic Oscillator

Cited by 36 publications

References 26 publications

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

New Quasi-Exactly Solvable Difference Equation

The classical hyperbolic Askey-Wilson dynamics without bound states

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