Eigenvalue asymptotics for the Schrödinger operator with a matrix potential in a single resonance domain

We consider a Schrodinger Operator with a matrix potential defined in L-2(m) (Gamma) by the di ff erential expression L(phi(x)) = (-Delta + V(x))phi(x) and the Neumann boundary condition, where F is the d dimensional rectangle and V is a martix potential, m >= 2, d >= 2. We obtain the asymptotic formulas of arbitrary order for the single resonance eigenvalues of the Schrodinger operator in L-2(m) (F).