Free Motion of Particles in the Lobachevskii Space in Terms of the Scattering Theory
DOI:
https://doi.org/10.15407/ujpe64.12.1108Keywords:
Lobachevskii space, horosphere, Fourier transform, coordinates, Schr¨odinger equation, scattering, amplitude of scattering, analytical representation, method, iterations, finite differencesAbstract
The problem of the motion of a free particle in the three-dimensional Lobachevskii space are interpreted as scattering by the space. The quantum-mechanical case is considered on the basis of the integral equation derived from the Schr¨odinger equation. After the separation of variables in a quasi-Cartesian coordinate system, the integral equation is derived for the momentum component along the axis of symmetry of a horosphere, which coincides with the z axis. The relationship between the scattering amplitude and analytical functions is established. The methods of iteration and finite differences are used to solve the integral equation.
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