Number of Bound States of Schrödinger Operators with Matrix-valued Potentials

We consider the Schrödinger operator −∆ − V (x) on R, but with the difference from the usual case that V is a Hermitian matrix-valued potential. In other words, the Hilbert space is not L(R) but L(R;C). The values of functions in this space, ψ(x), are N−dimensional vectors. (What we say here easily generalizes to ‘operatorvalued’ potentials, i.e., C is replaced by a Hilbert space such as L(R), but we stay with matrices in order to avoid technicalities.) The Cwikel-Lieb-Rozenblum (CLR) bound for d ≥ 3 in the scalar case N = 1 states that #(−∆ − V ), the number of negative eigenvalues of −∆− V , can be estimated by