We consider a Schrödinger Operator with a matrix potential defined in $L^m_2(F)$ by the differential expression \[ L(hi(x))=(-\Delta+V(x))hi(x) \] and the Neumann boundary condition, where $F$ is the $d$ dimensional rectangle and $V$ is a martix potential, $m\geqslant2,d\geqslant2$. We obtain the asymptotic formulas of arbitrary order for the single resonance eigenvalues of the Schrödinger operator in $L^m_2(F)$.