A certain model of formation of a fractal dimensionality in the Schrödinger problem
The main objective of the author is to investigate the behaviour of the integral curves of the following nonlinear system \[ (1)\quad \psi ''(x)+[\alpha q'(x)+E]\psi (x)=0,\quad (2)\quad q''(x)+a^ 2[1-\beta \psi^ 2(x)]q(x)=0 \] in a small vicinity of its special points. Here \(\alpha >0\), \(\beta >0\), \(a>0\) and E are known constants. This system constitutes a one-dimensional model for electrons which interact with acoustic phonons. If \(E\neq 0\), the special point consists of the unique trivial solution \((\psi_ 0=0\), \(q_ 0=0\), \(\psi_ 0'=0\), \(q_ 0'=0)\) and does not exhibit any difficulty. On the contrary, when \(E=0\), \(\psi_ 0\) remains arbitrary and thus causes some difficulties. In order to avoid these difficulties the author adds some small nonlinear terms to (1) and (2), namely: \(\eta \psi^ 2\) and \((-\sigma q^ 2)\), respectively. By studying then the modified system near its unique special point, the author reveals some properties of the trajectory.
scientific article; zbMATH DE number 3983660