I was wondering if anybody could give me some references to already
existing literature for the following open ended problem.


Namely, I am interested in studying the equation of
"complex harmonic oscillator"


$$\ddot{z}(t)+q(t)z(t)=0$$


where $z:\mathbb{R}\to\mathbb{C}$.


The case when $t$ is complex is also interesting and might shed some 
light on to the real case. Assume that the function $q(t)$ is real (for
real time) and strictly decreasing. Actually, without lost of generality
let $q(t)=1/(1+t)$. I have strong numerical evidence for the
following claim. There exist the set of initial conditions such that
$|z|$ is a constant or strictly monotone decreasing (increasing)
functions on the solutions satisfying those initial conditions.


I am considering the following auxiliary function
$\rho(t)=\ln(z\cdot \bar{z})$
the real case but I still can not make a use of the fact that q(t) is
positive and decreasing on $[0,\infty)$.
For the curious, I am essentially play the same kind a game like in the
proof of Sturm-Picone comparison theorem or Poincare-Benedixon theorem.


I started looking into the case of the complex time due to the following
simple observation. Suppose that $z\_1(t)$ and $z\_2(t)$ are two linearly
independent holomorphic solutions of the equation. The Schwarzian
derivative in notation $S$ of the ratio $z\_1/z\_2$ satisfies


$$S(\frac{z\_1}{z\_2})=1/(1+t)$$


Thank you.



Complex harmonic oscilator
The d.e. $z'' + z/(1+t) = 0$ is not at all general, in fact it has closed-form solutions $z \left( t \right) =c\_{{1}}\sqrt {1+t}\ 
 {J\_1 \left(2\sqrt {1+t}\right)}+c\_{{2}}\sqrt {1+t}\ 
 {Y\_1 \left(2\sqrt {1+t}\right)}$ where $J\_1$ and $Y\_1$ are the Bessel functions of the first kind and order 1. In fact your equation becomes Bessel's equation of order 1 under the transformation $ t = s^2/4 - 1,\ z(t) = s y(s)$.


The asymptotics of these functions are well known: $z(t) = - \frac{t^{1/4}}{\sqrt{\pi}}(c\_1 \cos(2 \sqrt{1+t} +\pi/4) + c\_2 \sin(2 \sqrt{1+t} + \pi/4)) + O(t^{-1/4})$. So your conjecture is false: all the real solutions (except the constant 0) oscillate with increasing amplitude as $t \to \infty$.


