Space with spinor structure and analytical properties of the solutions of Klein-Fock equation in cylindric parabolic coordinates

Possible quantum mechanical corollaries of changing the vectorial geometrical model of the physical space, extending it twice, in order to describe its spinor structure (in other terminology and emphasis it is known as the Hopf's bundle) are investigated. The extending procedure is realized in cylindrical parabolic coordinates: G(t, u, v, z) double right arrow (G) over tilde (t, u, v, z). It is done through expansion twice as much of the domain G so that instead of the half plane (u, v > 0) now the entire plane (u, v) should be used accompanied with new identification rules over the boundary points. Solutions of the Klein-Fock and Schrodinger equations Psi is an element of, p, a = e(i is an element of t)e(ipz) U-a(u)V-a(v) are constructed in terms of parabolic cylinder functions, a is a separating constant. Given quantum numbers c,p,a four types of solutions are possible: Psi(++), Psi(--); Psi(+-,) Psi(-+). The first two Psi(++),Psi(--) provide us with single-valued functions of the vectorial space points, whereas last two Psi(+-), Psi(-+) have discontinuities in the frame of vectorial space and therefore they must be rejected in this model. All four types of functions are continuous ones being regarded in the spinor space. It is shown that solutions Psi(++),Psi(--), Psi(+-), Psi(-+) all are the eigen-functions of two discrete spinor operators (delta) over cap and (pi) over cap: (delta) over cap (u, v) = ( -u, -v), (pi) over cap (u, v) = (u, -v), (delta) over cap (x, y) = (x, y), (pi) over cap (x, y) = (x, y). Two other classifications of the wave functions over discrete quantum numbers are given. It is established that all solutions Psi(++),Psi(--), Psi(+-), Psi(-+) are orthogonal to each other provided that integration is done over extended domain parameterizing the spinor space. Simple selection rules for matric elements of the vector and spinor coordinates, (x, y) and (u, v), respectively, are derived. Selection rules for (u, v) are substantially different in vector and spinor spaces. In the supplement some relationships describing primary geometric objects, spatial spinor xi and eta, as functions of cylindrical parabolic coordinates, are given.