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.