Shapiro's plane waves in spaces of constant curvature and separation of variables in real and complex coordinates

The aim of the article to clarify the status of Shapiro plane wave solutions of the Schr\"odinger'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 Schr\"odinger 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 Schr\"{o}dinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of Schr\"odinger equation in complex horosperical coordinates of $S_{3}$. In particular, two oppositely directed plane waves may be presented as exponentials in conjugated coordinates. $\Psi_{-}= e^{-\alpha z}$ and $\Psi_{+}= e^{-\alpha z^{*}}$. Solutions constructed are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels.