On bound states for systems of weakly coupled Schrödinger equations in one space dimension

We establish the Birman–Schwinger relation for a class of Schrödinger operators −d2/dx2⊗1H+V on L2(math,H), where H is an auxiliary Hilbert space and V is an operator-valued potential. As an application we give an asymptotic formula for the bound states which may arise for a weakly coupled Schrödinger operator with a matrix potential (having one or more thresholds). In addition, for a two-channel system with eigenvalues embedded in the continuous spectrum we show that, under a small perturbation, such eigenvalues turn into resonance