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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"


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


Thank you.

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Why do you say "without lost of generality" when you choose $q(t)$? Does that choice of $q(t)$ obey a universality property? –  S. Carnahan May 26 '11 at 5:43
There is a whole class of quantum mechanics systems and an interesting question about them which essentially boils down to the above question about that particular ODE. The value of the function $q(t)=1/(1+t)$ is a particular value for function $q(t)$ for one of those systems. There are even some in which $q(t)$ is a periodic function (Floquet case). So the phrase without loss of generality is really a bad choice but even with that particular value of $q(t)$ I will be able to answer the question at least about one interesting quantum mechanics problem. –  Predrag Punosevac May 26 '11 at 13:51
If you would like to read about the Schroedinger equation with a complex valued <i>polynomial</i> potential, I point you to my licentiate thesis: www2.math.su.se/reports/2010/5/2010-5.pdf The property about the Schwarzian derivative is mentioned there. There might be some inspiration there? –  Per Alexandersson May 26 '11 at 18:49
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2 Answers

up vote 4 down vote accepted

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 $t \to \infty$.

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I appreciate your answer very much! I will have to think more about it but I have a quick question. My understanding is that Bessel DE is real ODE. Note that the function $z(t)$ is complex valued even when $t$ is real. Actually the above equation easily decouples in the case of real $t$ (real valued function $q(t)$) into two Bessel ODEs. $$\ddot{x}+q(t)x=0$$ $$\ddot{y}+q(t)y=0$$ My claim is about $|z|=\sqrt{x^2+y^2}$. Showing that $|z|$ increasing is very, very GOOD! See above comment why. Could you give me a reference? –  Predrag Punosevac May 26 '11 at 14:32
The complex solutions with $c_1= \pm i c_2$ will have $\frac{∣z(t)∣^2}{∣c1|^2} =\frac{\sqrt{t}}{\pi}+\frac{19}{32 \pi \sqrt{t}} +O(t^{−3/2})$ and $\frac{d}{dt} \frac{|z|^2}{∣c1|^2} = \frac{1}{2 \pi \sqrt{t}} −\frac{19}{64 t^{3/2} \pi} +O(t^{−5/2})$ as $t \to + \infty$, and in particular $|z|$ will be increasing for sufficiently large real $t$. In fact, it appears to be increasing for all $t>−1$. I used Maple for all this, but the asymptotics of Bessel functions should be found in standard references such as Abramowitz and Stegun. –  Robert Israel May 26 '11 at 16:31
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Relevant references:

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I have both books on my shelf. –  Predrag Punosevac May 26 '11 at 13:33
I should be more careful before dismissing answers. For the past two-three days I have carefully read chapter five of the Hille's book at least dozen of times as well as skimming through several other chapters. The book contains exactly the kind of information I was craving for! –  Predrag Punosevac Jul 8 '11 at 18:10
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