arXiv2010
The aim of the article to clarify the status of Shapiro plane wave solutions of the Schrödinger's equation in the frames of the well-known general method of separation of variables. To solve this task, we use the well-known cylindrical coordinates in Riemann and Lobachevsky spaces, naturally related with Euler angle-parameters. Conclusion may be drawn: the general method of separation of variables embraces the all plane wave solutions; the plane waves in Lobachevsky and Riemann space consist of a small part of the whole set of basis wave functions of Schrödinger equation. In space of constant positive curvature $S_{3}$, a complex analog of horospherical coordinates of Lobachevsky space $H_{3}$ is introduced. To parameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obey additional restriction in the form of the equation $r^{2} = e^{z-z^{*}} - e^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$ with additional constraint, or through pairs of conjugate variables $(r,r^{*})$ or $(z,z^{*})$; correspondingly exist three different representations for Schrödinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of Schrödinger equation in complex horosperical coordinates of $S_{3}$. In particular, two oppositely directed plane waves may be presented as exponentials in conjugated coordinates. $Ψ_{-}= e^{-αz}$ and $Ψ_{+}= e^{-αz^{*}}$. Solutions constructed are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels.