  We consider a Schr\"odinger Operator with a matrix potential defined in
$L_2^m(F)$ by the differential expression\begin{equation*}
  L(\phi(x))=(-\Delta+V(x))\phi(x)
  \end{equation*}and the Neumann boundary condition, where $F$ is the $d$
dimensional rectangle and $V$ is a martix potential, $m\geqslant 2, d\geqslant
2$. We obtain the asymptotic formulas of arbitrary order for the single
resonance eigenvalues of the Schr\"odinger operator in $L_2^m(F)$.
