Uniqueness of a solution of a system of equations

Question

While studying quantummechanics, I encountered following algebraic problem: We know that if $l$ is a non-negative integer: $$2l+1 = \sum_{-l}^l{c_m(-1)^m}$$ $$2l+1 = \sum_{-l}^l{\vert c_m\vert^2}$$ Where $c_m$ are coefficients, that may be complex. Obviously $$ c_m = (-1)^m $$ Is a solution of this system of equations. I'm not sure though how I can see if this is the only solution. This is needed for proving the addition theorem of spherical harmonics.

Answer 1

Use the Cauchy-Schwarz inequality: $$ \sum_{m=-l}^l c_m(-1)^m \leq \sqrt{\textstyle\sum_{m=-l}^l \lvert c_m \rvert^2 \sum_{m=-l}^l 1} = 2l + 1. $$ Because the two sides are in fact equal, the vector $(c_m)_{m=-l}^l$ must be a scalar multiple of the vector $((-1)^m)_{m=-l}^l$. Denote this scalar factor by $a$. Then $\sum_{m=-l}^l c_m(-1)^m = \sum_{m=-l}^l a = (2l+1)a \implies a=1$.