The relativistic Hermite polynomial is a Gegenbauer polynomial

Nagel, Bengt

doi:10.1063/1.530606

Cited by 28 publications

(18 citation statements)

References 2 publications

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“…The RHP have been studied by different authors and related to other already known polynomials, such as Jacobi [24] or Gegenbauer [25] polynomials, and the essential of the latter is here collected since it is relevant for the next section. In fact, in [25] is proved the actual relation:…”

Section: The Example Of the Relativistic Harmonic Oscillatormentioning

confidence: 99%

“…[25] it is commented that "H N n (ξ) can actually be expressed directly as a (generalized) Gegenbauer polynomial in the form C…”

Section: The Example Of the Relativistic Harmonic Oscillatormentioning

confidence: 99%

“…[10] where the connection to the motion in a homogeneous space under the group SO(1, 2), that is, the Antide Sitter universe, is studied. We use the notationĈϕ to hilight that the l.h.s in (25) is the quantum realization of the Casimir operator of the Lie algebra of SL(2, R) [16].…”

Section: The Example Of the Relativistic Harmonic Oscillatormentioning

confidence: 99%

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Group approach to the quantization of the Pöschl–Teller dynamics

Aldaya¹,

Guerrero

2005

J. Phys. A: Math. Gen.

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The quantum dynamics of a particle in the Modified Pöschl-Teller potential is derived from the group SL(2, R) by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2, R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU (2) where all the states are physical.

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Section: The Example Of the Relativistic Harmonic Oscillatormentioning

confidence: 99%

“…[25] it is commented that "H N n (ξ) can actually be expressed directly as a (generalized) Gegenbauer polynomial in the form C…”

Section: The Example Of the Relativistic Harmonic Oscillatormentioning

confidence: 99%

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Group approach to the quantization of the Pöschl–Teller dynamics

Aldaya¹,

Guerrero

2005

J. Phys. A: Math. Gen.

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“…+ n and H (N,γ) n (χ) are polynomials in the variable χ ≡ mω h x (see [22,20] for more details). The invariant integration volume inG defining the scalar product can be written as:…”

Section: Configuration-space Imagementioning

confidence: 99%

Untitled

Calixto¹

2000

Int. J. Mod. Phy. A

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In this article we propose a "second quantization" scheme especially suitable to deal with non-trivial phase spaces, implemented within a more general Group Approach to Quantization, which recovers the standard Quantum Field Theory (QFT) for ordinary relativistic linear fields. We emphasize, among its main virtues, the capability of selecting a globally defined vacuum in a QFT on a curved space-time and the absence of the usual requirement of global hyperbolicity of space-time. We illustrate our QFT approach with the example of the Anti-de Sitter universe.PAC numbers: 03.65.Fd, 04.62.+v 1

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“…Notably, both the RHPs and the RLPs are not independent polynomials, being indeed proved to relate to the Jacobi polynomials of suitable parameters and arguments [27][28][29]; specifically, the RHPs relate to the Gegenbauer polynomials. However, since their formal expressions allow for a direct analogy with the Hermite-Gaussian and Laguerre-Gaussian solutions to the wave equations, we prefer to use such formal expressions and to refer to them in accord with the original terminology, which has also the advantage of evoking the "relativistic" context where those expressions naturally frame.…”

Section: Introductionmentioning

confidence: 99%

The Relativistic Hermite Polynomials and the Wave Equation

Torre¹

2009

PIER B

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Abstract-Solutions of the homogeneous 2D scalar wave equation of a type reminiscent of the "splash pulse" waveform are investigated in some detail. In particular, it is shown that the "higher-order" solutions relative to a given "fundamental" one, from which they are obtained through a definite "generation scheme", come to involve the relativistic Hermite polynomials. This parallels the results of a previous work, where solutions of the 3D wave equation involving the relativistic Laguerre polynomials have been suggested. Then, exploiting a well known rule, the obtained wave functions are used to construct further solutions of the 3D wave equation. The link of the resulting wave functions with those analyzed in the previous work is clarified, the pertinent generation scheme being indeed inferred. Finally, solutions of the Klein-Gordon equation which relate to such Lorentzian-like solutions of the scalar wave equation are deduced.

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The relativistic Hermite polynomial is a Gegenbauer polynomial

Cited by 28 publications

References 2 publications

Group approach to the quantization of the Pöschl–Teller dynamics

Group approach to the quantization of the Pöschl–Teller dynamics

Untitled

The Relativistic Hermite Polynomials and the Wave Equation

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