Uniqueness of Ground States for Schrِdinger Operators over Loop Groups

Over the space L of based loops in a compact connected Lie group G there is a natural gradient operator ∇ induced by the left action on L of the Hilbert Lie group K0 which consists of finite energy elements of L. Denote by μ pinned Brownian motion measure on L. The Schrödinger operator ∇ + ∇ + V acting in L2(L, μ) is shown to have a unique ground state over each homotopy class in L. The proof of uniqueness is reduced to the proof of ergodicity of the left action of K0 on (L, μ) for a simply connected group G of compact type. Ergodicity, in turn, is proved by first characterizing the Itô expansion coefficients of K0 invariant functions and then regularizing such functions via a smooth approximation of multiple Stratonovich integrals. As a byproduct of this method one obtains an isometric isomorphism of L2(G, heat kernel measure at time one) with a natural completion of the universal enveloping algebra of G, thus extending to the noncommutative case the well known isometry between L2(Rd, Gauss measure) and the space of symmetric tensors over Rd.