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Shapiro’s plane waves in spaces of constant curvature and separation of variables in real and complex coordinates . (English) Zbl 1175.35134

Summary: The goal of the paper is 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 the Schrödinger equation. In space of constant positive curvature \(S_{3}\), a complex analog of horispherical 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 the 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 there exist three different representations for the Schrödinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of the Schrödinger equation in complex horisperical 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
76Q05 Hydro- and aero-acoustics
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