Node Theorem for Matrix Schroedinger Operators

TL;DR

This paper generalizes the classical node theorem to matrix Schrödinger operators, establishing conditions under which ground states have no zeros and analyzing their degeneracy using path integrals and cluster expansion techniques.

Contribution

It extends the classical node theorem to matrix Schrödinger operators and introduces a novel proof approach using path integrals and cluster expansion.

Findings

Ground states have no zeros under specified conditions.

Degeneracy of ground states is at most m.

Provides a new proof method for the node theorem using path integrals.

Abstract

In this paper we study the ground states of a matrix Schroedinger operator, that is an operator of the type (-Laplace) + V acting on m-component wave functions in R^n. We prove in generalization of the classical node theorem that the ground states of this operator have no zeros, if the potential V satisfies the following conditions: 1) V is bounded from below. 2) V is strictly positive at infinity with respect to the ground state energy. 3) The partial derivatives of V do not grow too fast in the order of the derivative. Furthermore we show that the degeneracy of the ground states is at most m. For the proof we reformulate the problem with path integrals and perform a cluster expansion with large/small field conditions.