Logarithmic and hyperbolic spirals associated with Schrödinger's equation
type: Item
instance of: scholarly article
title: Logarithmic and hyperbolic spirals associated with Schrödinger's equation
zbMATH Open document ID: 1497.34003
published in: Journal of Differential Equations
publication date: 2022-02-10T00:00:00Z
review text: Geometric properties of the solutions of the equation \[ Q''+\frac{2}{\zeta}Q'-Q+ia(\zeta Q)'=0, \quad 0<\zeta<\infty \] in the \((\mathrm{Re}(Q),\mathrm{Im}(Q))\)-plane are investigated, where \(a>0\) is a parameter. Assuming \(Q\) in polar form \(Q = \rho e^{i\theta}\), a coupled nonlinear system is derived and investigated. Three diverse solutions of this system are analyzed. The first two, \(Q_j = \rho_j e^{i\theta_1}\), \(j=1,2\) exist \(\forall \zeta>0\), and become singular as \(\zeta \to 0^+\) They rotate around \((0, 0)\) with \({\theta}'_j(\zeta)<0\), \(j=1,2\) and form logarithmic and hyperbolic spirals when \(\zeta \gg 1\). The third solution, \(Q_3 = \rho_3 e^{i\theta_3}\) has the properties \(0<\lim_{\zeta\to 0^+}\rho_3(\zeta)<\infty\), \({\theta}'_3(\zeta)\) changes sign infinitely often on \(0 < \zeta <\infty)\), causing the formation of ``internal oscillations'' in the trajectory of the solution.
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 7471741
zbMATH Keywords: topological shooting
zbMATH Keywords: logarithmic spiral
zbMATH Keywords: hyperbolic spiral
author: W. C. Troy
MaRDI profile type: Publication
full work available at URL: j.jde.2022.01.005
OpenAlex ID: W4205801846
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DOI: 10.1016/J.JDE.2022.01.005
description: scientific article; zbMATH DE number 7471741