Logarithmic and hyperbolic spirals associated with Schrödinger's equation
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.
scientific article; zbMATH DE number 7471741