The motion of the classical and quntum partcles in the extended Lobachevsky space

1. The motion of the classical and quntum partcles in the extended Lobachevskyspace Yu. Kurochkin, V.S. Otchik, E. Ovseyuk, Dz. Shoukovy

2. Plan Introduction Classical problem Quantum problem Perspectives

3. Introduction • Quantum-mechanical problems in the spaces of a constant positive and negative curvature are the object of interest of researchers since 1940, when Schrödinger was first solved the quantum-mechanical problem about the atom on the three-dimensional sphere S3. The analogous problem in the three-dimensional Lobachevsky space 1S3 was first solved by Infeld and Shild and imaginary Lobachevsky space C. Grosche (1994). These authors found the energy spectrum to be degenerate similarly to that in flat space. • In recent years the quantum-mechanical models based on the geometry of spaces of constant curvature have attracted considerable attention due to their interesting mathematical features as well as the possibility of applications to physical problems

4. Introduction • An additional constant of motion, analog of the Runge-Lenz vector for the problem on the sphere S3 and for Lobachevsky space 1S3 togetherwith angular momentum generate algebraic structure which may be considered as a nonlinear extension of Lie algebra, and which was called cubic algebra [1,2,3,4,5]. • Kepler-Coulomb problem on the sphere S3 has been used as a model for description of quarkonium spectrum, and ecxitons semiconductor quantum dots [6] . [1]P. Higgs// J. Phys A. Math. Gen., 12, 309, (1979) [2] H. Leemon J. Phys A. Math. Gen., 12 , 489, (1979) [3]Yu. Kurochkin, V. Otchik// Dokl. Akad. Nauk BSSR, 23, (1979) [4] A. Bogush, Yu. Kurochkin, V. Otchik// Dokl. Akad. Nauk BSSR, 24, (1980) [5] A. Bogush, Yu. Kurochkin, V. Otchik// ЯФ, 61, (1998) [6] V. Gritzev, Yu. Kurochkin// Phys. Rev B, 64, (2001)

5. The interpretation of the three dimensional extended Lobachevsky space in terms of three dimensional Euclidean space As is well known there exist interpretations (F. Klein, E. Beltrami) of the three dimensional spaces of constant curvature in terms of three dimensional Euclidean spaces. These interpretations provide in particular applications of the quantum mechanical models based on the geometry of the spaces of constant curvature to the solution of some problems in the flat space. For example the following interpretation of the three dimensional Lobachevsky space can be used: 1. Real three dimensional Lobachevsky space inside of three dimensional sphere of three dimensional Euclidean space (1) 2. Imaginary three dimensional Lobachevsky space outside of three dimensional sphere of three dimensional Euclidean space Here are coordinates of points in the three - dimensional Euclidean space (2) R - radius of sphere in the Euclidean space and radius of curvature in the Lobachevsky real and imaginary spaces in the realization defined by formulas (1),(2)

6. Spherical coordinates for the real Lobachevsky space Spherical coordinates for the imaginary Lobachevsky space Metrical tensor of the real Lobachevsky space

7. Free particle (real Lobachevsky space) Hamilton – Jacoby equation where Solution

8. Free particle (imaginary Lobachevsky space) Metrical tensor of the imaginary Lobachevsky space Hamilton – Jacoby equation where Solution

9. Coulomb potential Real space. Hamilton – Jacoby equation Solution where .

10. Coulomb potential Imaginary space. Hamilton – Jacoby equation Solution where

11. A charged particle in the constant homogeneous magnetic field in the extended Lobachevsky space. Real space Metrical tensor is Hamilton – Jacoby equation Solution where .

12. A charged particle in the constant homogeneous magnetic field in the extended Lobachevsky space. Imaginary space Metrical tensor is Hamilton – Jacoby equation Solution , where .

13. QUANTU MECHANICAL PROBLRM The Schrödinger equation for Kepler-Coulomb problem on the sphere S3 and in the Lobachevsky space 1S3 is where xµarecoordinates in four-dimensional flat space. R is a radius of the curvature; for 1S3 R= iρ With Hamiltonian commute angular momentum operator And analog Runge-Lenz operator , where

14. QUANTU MECHANICAL PROBLEM Operators Aiand Li obey the following commutation relation The energy spectra of the Hamiltonians are  S3 space; n is the principal quantum number 1S3 space

15. Gelfand-Graev transformation of the wave function in the real Lobachevsky space Here The inverse formula where - measure on the Lobachevsky space. The analog plane wave , is the solution of the Schrodinger equation when

16. Gelfand-Graev transformation of the wave function in the imaginary Lobachevsky space In the imaginary space The inverse formulas to isotropic direct line - is distinction from point ,

17. Quantum mechanical problem. Coulomb potential • Parabolic coordinates in the Lobachevsky space Parabolic coordinates In the imaginary Lobachevsky space

18. Solutions of the Schroedinger equation in imaginary Lobachevsky space • Substitution separates the variables and equations for and in the case of imaginary Lobachevsky space are where separation constantsand obey the relation Solutions of these equations can be expressed in terms of hypergeometric functions Here we have introduced the notations