arXiv 1001.53782010-01-29math-phmath.MP

常曲率空间中的Shapiro平面波和实坐标与复坐标中的分离变量

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

E. M. Ovsiyuk, N. G. Tokarevskaya, V. M. Red'kov

备注：: 19 pages
发表信息：: Nonlinear Phenomena in Complex Systems. 12, 1--15 (2009)

摘要：

本文的目的是阐明薛定谔方程的Shapiro平面波解在著名的分离变量法框架内的地位。为了解决这个问题，我们使用著名的圆柱坐标在黎曼和罗巴切夫斯基空间，自然与欧拉角参数。可以得出结论：一般的分离变量法包含了所有的平面波解，Lobachevsky空间和Riemann空间中的平面波只包含了薛定谔方程全部基波函数的一小部分。在常正曲率空间$S_{3}$中，引入了Lobachevsky空间$H_{3}$的准球坐标的复模拟.为了参数化实空间$S_{3}$，两个复坐标$(r,z)$必须服从以方程$r^{2} = e^{z-z^{*}} - e^{2z} $形式的附加约束。空间$S_{3}$的度量张量可以用附加约束的$(r,z)$表示，或者通过共轭变量对$(r,r^{*})$或$(z,z^{*})$表示;相应地，薛定谔哈密顿量存在三种不同的表示。确定了Shapiro平面波作为薛定谔方程在$S_{3}$复宇宙坐标系中的解。特别地，两个相反方向的平面波可以被呈现为共轭坐标中的指数。$Ψ_{-}= e^{-αz}$和$Ψ_{+}= e^{-αz^{*}}$。构造的解决方案是单值的，有限的，在球形空间中的连续函数，并对应于离散的能量水平。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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