Bicomplex Harmonic and Isotonic Oscillators: The Excited States

Banerjee, Abhijit

doi:10.1007/s00006-017-0772-4

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…Since then many applications of bicomplex and hyperbolic numbers have been published in mathematics and physics. Only a few of them are listed here [83,84,85,86,87,88,89,90,91,92,93,94,95,96,97]. Further references can be found in these articles.…”

Section: Bicomplex Numbersmentioning

confidence: 99%

Holographic coordinates

Ulrych¹

2018

Journal of Mathematical Analysis and Applications

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The Laplace equation in the two-dimensional Euclidean plane is considered in the context of the inverse stereographic projection. The Lie algebra of the conformal group as the symmetry group of the Laplace equation can be represented solely in terms of the solutions and derivatives of the solutions of the Laplace equation. It is then possible to put contents from differential geometry and quantum systems, like the Hopf bundle, relativistic spin, bicomplex numbers, and the Fock space into a common context. The basis elements of the complex numbers, considered as a Clifford paravector algebra, are reinterpreted as differential tangent vectors referring to dilations and rotations. In relation to this a homogeneous space is defined with the Lie algebra of the conformal group, where dilations and rotations are the coset representatives. Potential applications in physics are discussed. et al. [25]. Furthermore, a hierarchy of projective geometric spaces and Möbius geometries can be introduced based on Vahlen matrices [26,27,28,29]. Maks [30] investigated explicitly the sequence of Möbius geometries represented in terms of Clifford algebras, which is considered also in [31] in a bicomplex matrix representation. In relation to particle physics one may assume that projections of a more general geometry can arise as part of the measurement process via electromagnetic forces. In accordance with the holographic principle Möbius geometries and Möbius transformations can provide here a new perspective on the experimental data.Atiyah, Manton, and Schroers [32,33,34,35] describe electrons, protons, neutrinos, and neutrons within a model that has been inspired by Skyrme's baryon theory [36]. The Skyrme model became popular when Witten came up with the idea that baryons arise as solitons of the classical meson fields [37]. The Skyrme theory turned out to be remarkably successful in describing baryons in the previous decades [38]. Thus matter is supposed to arise from a pure geometric foundation [39,40]. Solitons are usually considered in the context of gauge transformations, which relate in a mathematical context to fiber bundles and Cartan geometries [22].With regards to the topics discussed above research in two-dimensional conformal field theories [41,42,43,44,45] has turned out to be important, also because conceptual insights can be obtained more easily in a lower dimensional geometry. Motivation to proceed in this direction is also provided by [31]. In this article the two-dimensional Euclidean plane and the conformal symmetries, which refer to the Laplace equation in R 2 , are fundamental in a sequence of higher dimensional geometries and Clifford algebras. Therefore, one can ask the question how the Laplace equation can be embedded into higher dimensional geometries. Here concepts like the stereographic projection should play an important role. In this sense one can consider the Laplace equation in the base space S 2 instead of R 2 by means of what is denoted here as holographic coordinates. This conceptual change is adopt...

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Section: Bicomplex Numbersmentioning

confidence: 99%

Holographic coordinates

Ulrych¹

2018

Journal of Mathematical Analysis and Applications

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“…The theory of bicomplex numbers is a matter of active research in recent times, and there are many further discussions in this direction. Banerjee studied bicomplex Fourier transform, 8 bicomplex Laplace transform, 9 bicomplex indefinite inner product modules, 10 and bicomplex Harmonic and Isotonic Oscillators 11 . Agarwal discussed bicomplex Maxwell's equations, 7 bicomplex Mittag–Leffler function, 12 and bicomplex polygamma function 13 .…”

Section: Introductionmentioning

confidence: 99%

“…Banerjee studied bicomplex Fourier transform, 8 bicomplex Laplace transform, 9 bicomplex indefinite inner product modules, 10 and bicomplex Harmonic and Isotonic Oscillators. 11 Agarwal discussed bicomplex Maxwell's equations, 7 bicomplex Mittag-Leffler function, 12 and bicomplex polygamma function. 13 In addition, bicomplex Riemann zeta function, finite and infinite dimensional bicomplex Hilbert spaces, bicomplex Hardy space, and bicomplex quantum mechanics were discussed in earlier studies.…”

Section: Introductionmentioning

confidence: 99%

The Schwarz lemma in bicomplex analysis

Dai

2023

Math Methods in App Sciences

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We investigate Schwarz lemma in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. The bicomplex is a generalization of complex which has close relations with Fractal geometry, Minkowski Space-Time, Maxwell's equations, Schrödinger equation, and Gaussian pulse wave. In this paper, we first construct a type of bicomplex Möbius transformation and obtain some results: the mapping properties on bicomplex sphere and bicomplex ball, preserving the inverse points with respect to the bicomplex sphere B(0, 1). Then we obtain the Poisson integral formula in bicomplex setting, and by using the Poisson integral formula, we give the Schwarz lemma for bicomplex holomorphic functions in bicomplex setting.Finally, we shall give the Schwarz lemma and the Schwarz-Pick type lemma for holomorphic functions in bicomplex analysis. These results may give new energy for the development of quantum mechanics.

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“…Hyperbolic extensions of the complex Hilbert space have been studied in [9]. The standard Schrödinger equation was bicomplexified in [10] and further studied in [11,12,13,14]. Quaternionic and coquaternionic quantum mechanics and quantum field theory have been studied for a long time, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Multicomplex solitons

Cen¹,

Fring²

2019

JNMP

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We discuss integrable extensions of real nonlinear wave equations with multisoliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton solutions with various types of novel qualitative behaviour can be constructed. Corresponding to the different multicomplex units in these extensions, real, hyperbolic or imaginary, the wave equations and their solutions exhibit multiple versions of antilinear or PT -symmetries. Utilizing these symmetries forces certain components of the conserved quantities to vanish, so that one may enforce them to be real. We find that symmetrizing the noncommutative equations is equivalent to imposing a PT -symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.

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Bicomplex Harmonic and Isotonic Oscillators: The Excited States

Cited by 5 publications

References 15 publications

Holographic coordinates

Holographic coordinates

The Schwarz lemma in bicomplex analysis

Multicomplex solitons

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