A new algebraization of the Laméequation

Finkel, Federico; González‐López, A.; Rodrı́guez, Miguel Á.

doi:10.1088/0305-4470/33/8/303

Cited by 31 publications

(46 citation statements)

References 22 publications

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“…All these solutions had the feature that in the uncoupled limit, they reduce to the well known solutions of the corresponding uncoupled φ 6 problem. The purpose of this paper is to point out that this coupled model has truly novel solutions in terms of Lamé polynomials of order one and two [7,8], provided we add (symmetry allowed) quartic-quadratic and quadratic-quartic couplings to the coupled φ 6 model considered earlier [6]. It may be noted that these solutions exist only because of the coupling between the two fields (up to sixth order).…”

Section: Introductionmentioning

confidence: 99%

Domain wall and periodic solutions of a coupled ϕ6 model

Khare

Saxena

2008

Journal of Mathematical Physics

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Abstract:We obtain several higher order periodic solutions of a Coupled φ 6 model in terms of Lamé polynomials of order one and two. These solutions are unusual in the sense that while they are the solutions of the coupled problem, they are not the solutions of the uncoupled problem. We also obtain exact solutions of coupled φ 6 -φ 4 models, both when the φ 4 potential corresponds to a first order (asymmetric double well) or a second order (symmetric double well) transition.

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Section: Introductionmentioning

confidence: 99%

Domain wall and periodic solutions of a coupled ϕ6 model

Khare

Saxena

2008

Journal of Mathematical Physics

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“…It plays nowadays a prominent role in physics appearing in such diverse theories as crystals models in solid state physics [6,7], exactly and quasi-exactly solvable quantum systems [8,9], integrable systems and solitons [10,11] [21], and preheating after inflation modern theories [22]. Most often, the Lamé equation appears in physics literature in the Jacobian form of a one-dimensional Schrodinger equation with a doubly periodic potential,…”

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confidence: 99%

Hidden nonlinear supersymmetry of finite-gap Lamé equation

2007

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A bosonized nonlinear (polynomial) supersymmetry is revealed as a hidden symmetry of the finite-gap Lamé equation. This gives a natural explanation for peculiar properties of the periodic quantum system underlying diverse models and mechanisms in field theory, nonlinear wave physics, cosmology and condensed matter physics. PACS numbers: 11.30.Pb; 11.30.Na; 03.65.Fd Supersymmetry [1], as a fundamental symmetry providing a natural mechanism for unification of gravity with electromagnetic, strong and weak interactions, still waits for experimental confirmation. On the other hand, in nuclear physics supersymmetry was predicted theoretically [2] and has been confirmed experimentally [3] as a dynamic symmetry linking properties of some bosonic and fermionic nuclei. It would be interesting to look for some physical systems whose special properties could be explained by a hidden ordinary (not a dynamic) supersymmetry.In the present Letter we show that the quantum system described by the finite-gap Lamé equation possesses a hidden supersymmetry. A very unusual nature of the revealed supersymmetry is that it manifests as a nonlinear symmetry of a bosonic system without fermion (spin) degrees of freedom. This means that we find here a kind of a bosonized supersymmetry giving a natural explanation for peculiar properties of the periodic quantum problem underlying many physical systems.The Lamé equation first arose in solution of the Laplace equation by separation of variables in ellipsoidal coordinates [4], and one of its early applications was in the quantum Euler top problem [5]. It plays nowadays a prominent role in physics appearing in such diverse theories as crystals models in solid state physics [6,7], exactly and quasi-exactly solvable quantum systems [8,9], integrable systems and solitons [10,11] [21], and preheating after inflation modern theories [22]. Most often, the Lamé equation appears in physics literature in the Jacobian form of a one-dimensional Schrodinger equation with a doubly periodic potential,where sn(x, k) ≡ sn x is the Jacobi elliptic odd function with modulus k (0 < k < 1), and real and imaginary periods 4K and 2iK ′ , K = K(k) is a complete elliptic integral of the first kind, and 4,23]. A remarkable property of this equation is that at integer values of the parameter j = n, its energy spectrum has exactly n gaps, which separate the n+1 allowed energy bands. The 2n + 1 eigenfunctions associated to the boundaries E i (n), i = 0, 1, . . . , 2n, of the allowed energy bands, ∞] are given by polynomials ('Lamé polynomials') of degree n in the elliptic functions sn x, cn x and dn x. These polynomials have real periods 4K or 2K, and the boundary energy levels E i (n) are non-degenerated. The states in the interior of allowed zones are described by the quasi-periodic Bloch-Floquet wave functions (which can be expressed in terms of theta functions [4]) of quasi-momentum κ(E),Every such interior energy level is doubly degenerated. For any non-integer value of the parameter j, Eq. (1) has an infinite num...

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“…Notice, however, that if there is no need to recover p 2 for all p ∈ R, any potential term that has a quadratic minimum at p = 0 of the same type as the one in the potential that appears in the Mathieu equation provides a perfectly valid approximation for the harmonic oscillator Hamiltonian at small energies. An interesting example in this respect is provided by periodic potentials defined by Jacobian elliptic functions, in particular those leading to the Lamé equation [22]. It is known that, in those cases the spectrum consists of a continuum of positive eigenvalues with a finite number of gaps (or, equivalently, on a finite number of bounded bands and an unbounded band).…”

Section: Actually the Width Of The Gaps Behaves As [21]mentioning

confidence: 99%

Band structure in the polymer quantization of the harmonic oscillator

Prieto

Villaseñor

2013

Class. Quantum Grav.

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We discuss the detailed structure of the spectrum of the Hamiltonian for the polymerized harmonic oscillator and compare it with the spectrum in the standard quantization. As we will see the non-separability of the Hilbert space implies that the point spectrum consists of bands similar to the ones appearing in the treatment of periodic potentials. This feature of the spectrum of the polymeric harmonic oscillator may be relevant for the discussion of the polymer quantization of the scalar field and may have interesting consequences for the statistical mechanics of these models.

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A new algebraization of the Laméequation

Cited by 31 publications

References 22 publications

Domain wall and periodic solutions of a coupled ϕ6 model

Domain wall and periodic solutions of a coupled ϕ6 model

Hidden nonlinear supersymmetry of finite-gap Lamé equation

Band structure in the polymer quantization of the harmonic oscillator

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