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What is the simplest solution for this set of equations: $ \sum_{i=1,3,5,..}^{N-1} \left | x_i \right |^2=c_1,\ \sum_{i=2,4,6,..}^{N} \left | x_i \right |^2=c_2,\ \sum_{i=1,3,5,..}^{N-1} x_i {x_{i+1}}^{*} =c_3 $ where $x_i$ are in general complex and $c_i$ are constants. If no simple solution exists, do you know an algorithm to obtain one numerically?

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The system seems underdetermined to me... –  J. M. Dec 11 '11 at 16:26
    
It is. So I am seeking an arbitrary solution. –  Tarek Dec 11 '11 at 17:14

1 Answer 1

up vote 1 down vote accepted

I understand from your question that $N$ is even and $x^*$ represents the complex conjugate of $x$, and I will asume this in the following, as well as $c_1>0$, $c_2>0$ and $N\ge4$.

The Cauchy–Schwarz inequality implies that a necessary condition for the existence of a solution is that $|c_3|^2\le c_1\,c_2$. Under this condition, it is easy to obtain solutions like the folowing: $x_i=0$ for $i\ge4$, and for $0\le\theta,\phi<2\,\pi$, $$ x_2=\sqrt{c_2}\,e^{i\theta},\quad x_1=\frac{c_3}{\sqrt{c_2}}\,e^{i\theta},\quad x_3=\sqrt{\frac{c_1\,c_2-|c_3|^2}{c_2}}\,e^{i\phi}\ . $$

Wether it is simple is up to you.

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