Node Theorem for Matrix Schrödinger Operators

In generalization of the classical node theorem we prove that the ground states of the matrix Schrödinger operator H = − 1 2 ∆ + V (x) , V ∈ C(IR, I C) have no zeroes, if the potential satisfies the following conditions: (I) V is bounded from below. (II) V − λ is strictly positive at infinity, where λ is the ground state energy. (III) The partial derivatives of V do not grow too fast in the order of the derivative. Furthermore we show that H has at most m ground states. For the proof we reformulate the problem with path integrals and perform a cluster expansion with large/small field conditions.