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We study here the behavior of the solutions to a $2\times 2$ semi-linearcooperative system involving Schr\" odinger operators (considered in itsvariational form): $$LU:=(-\Delta + q(x))U = AU+\mu U + F(x,U) \quad{\rm in}\\mathbb{R}^N$$ $$U(x)_{|x|\rightarrow \infty} \rightarrow 0$$ where $q$ is acontinuous positive potential tending to $+\infty$ at infinity; $\mu$ is a realparameter varying near the principal eigenvalue of the system; $U$ is a columnvector with components $u_1$ and $u_2$ and $A$ is a square cooperative matrixwith constant coefficient. $F$ is a column vector with components $f_1$ and$f_2$ depending eventually on $U$.

Semi-linear cooperative elliptic systems involving Schr{\"o}dinger operators: Groundstate positivity or negativity