§28.17 Stability as $x\to \pm \mathrm{\infty }$

If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. All other pairs are unstable.

For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$.

However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. For example, as $x\to +\mathrm{\infty }$ one of the solutions ${\mathrm{me}}_{\nu }(x,q)$ and ${\mathrm{me}}_{\nu }(-x,q)$ tends to $0$ and the other is unbounded (compare Figure 28.13.5). Also, all nontrivial solutions of (28.2.1) are unbounded on $ℝ$.

For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves $a=a_n\left(q\right)$ and $a=b_n\left(q\right)$; compare Figure 28.2.1.