Integrable systems of group SO(1,2) and Green’s functions

Dane, C.; Verdiyev, Y. A.

doi:10.1063/1.531375

Cited by 13 publications

(10 citation statements)

References 3 publications

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“…) and the phase factor (−1) (3.16b) as well as some typo in the formula referring to (3.16b) the solution (3.16) was guessed from (2.21) by R aczka, Limić and Niederle [11] and rediscovered by Dane and Verdiyev [4]. More precisely, one can put…”

Section: Pseudospherical Functions For Discrete Series Representationsmentioning

confidence: 99%

“…Both approaches are based on the very general mathematical scheme involving multidimensional hyperboloids, no wonder that the physically interesting case of the two-dimensional hyperboloid of one sheet was not discussed therein in a more detail. Recently, the wavefunctions for the two-dimensional hyperboloids involving their different parametrizations were studied in [4]. The approach used in [4] for the identification of wavefunctions in the hyperbolic parametrization of hyperboloids coincide with that adopted in [11].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the wavefunctions for the two-dimensional hyperboloids involving their different parametrizations were studied in [4]. The approach used in [4] for the identification of wavefunctions in the hyperbolic parametrization of hyperboloids coincide with that adopted in [11].…”

Section: Introductionmentioning

confidence: 99%

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Pseudospherical functions on a hyperboloid of one sheet

Kowalski¹,

Rembieliński²,

Szcześniak³

2011

J. Phys. A: Math. Theor.

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The pseudospherical functions on one-sheet, two-dimensional hyperboloid are discussed. The simplest method of construction of these functions is introduced using the Fock space structure of the representation space of the su(1, 1) algebra. The pseudospherical functions with half-integer order are investigated. The counterparts of the Legendre functions for the hyperboloid are introduced and a new class of pseudospherical functions is found.

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Section: Pseudospherical Functions For Discrete Series Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pseudospherical functions on a hyperboloid of one sheet

Kowalski¹,

Rembieliński²,

Szcześniak³

2011

J. Phys. A: Math. Theor.

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“…Then, the potential would offer an attractive force on the particle to form the bound states. For the H 2 surface, it was supposed to only have the scattering states (e.g., [25]). To solve the equation, introduce the new variablex = − sinh 2ȳ turning (A44) intō…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

The Magnetic Hooke-Newton Transmutation in Momentum Space

Lin

2021

Symmetry

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The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

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“…We also should mention works [36,44] where group SO(2, 2) was considered. More recently the wave functions on the two-dimensional one-sheeted hyperboloid in subgroup parametrization were presented in [12,46,71].…”

Section: Introductionmentioning

confidence: 99%

Lie-algebra contractions and separation of variables. Three-dimensional sphere

Pogosyan

Yakhno

2009

Phys. Atom. Nuclei

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In this work the detailed geometrical description of all possible orthogonal and nonorthogonal systems of coordinates, which allow separation of variables of two-dimensional Helmholtz equation is given as for two-sheeted (upper sheet) H2, either for one-sheetedH2 hyperboloids. It was proven that only five types of orthogonal systems of coordinates, namely: pseudo-spherical, equidistant, horiciclic, elliptic-parabolic and elliptic system cover one-sheetedH2 hyperboloid completely. For other systems onH2 hyperboloid, well defined Inönü-Wigner contraction into pseudo-euclidean plane E1,1 does not exist. Nevertheless, we have found the relation between all nine orthogonal and three nonorthogonal separable systems of coordinates on the one-sheeted hyperboloid and eight orthogonal plus three nonorthogonal ones on pseudo-euclidean plane E1,1. We could not identify the counterpart of parabolic coordinate of type II on E1,1 among the nine separable coordinates on hyperboloidH2, but we have defined one possible candidate having such a property in the contraction limit. In the light of contraction limit we have understood the origin of the existence of an additional invariant operator which does not correspond to any separation system of coordinates for the Helmholtz equation on pseudo-euclidean plane E1,1.Finally we have reexamine all contraction limits from the nine separable systems on two-sheeted H2 hyperboloid to Euclidean plane E2 and found out some previously unreported transitions.

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Integrable systems of group SO(1,2) and Green’s functions

Cited by 13 publications

References 3 publications

Pseudospherical functions on a hyperboloid of one sheet

Pseudospherical functions on a hyperboloid of one sheet

The Magnetic Hooke-Newton Transmutation in Momentum Space

Lie-algebra contractions and separation of variables. Three-dimensional sphere

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