Log-Sobolev inequalities with potential functions on pinned path groups

Abstract

We establish a refined version of Gross's log-Sobolev inequalities on pinned path groups. We explain the reason why it is useful in a lower bound estimate of Schrodinger operators on path spaces. semi-group and the equivalent notion of the validity of the log-Sobolev inequality are used to give a lower bound on the bottom of spectrum of a Schrodinger operator which is given by the sum of the generator of the semi-group and a potential function. We ap- plied this lower bound estimate to study the semi-classical behavior of the bottom of spectrum of Schrodinger operators on path spaces over compact Riemannian manifolds ((3, 4)) partly motivated by an application to P(`)-type Hamiltonian and an extension of (19) to infinite dimensional curved spaces. In this paper, we establish a refined version of Gross's log-Sobolev inequalities on a pinned path group. Pinned path group Pe,a(G) is a space of continuous paths with values in a compact Lie group G over the time interval (0,1) with a fixed starting point e (unit element) and the fixed end point a. We will apply the log- Sobolev inequalities with potential functions to study the semi-classical behavior of the low lying spectrum of Schrodinger operators on pinned path groups in a separate paper. The structure of the paper is as follows. In Section 2, we consider smooth pinned path spaces over a general compact Riemannian manifold M. We intro- duce a Riemannian structure on the pinned path space using a metric connection on M. Next we calculate the gradient of the energy function E(∞) = 1 R 1 0 |˙ ∞(t)| 2 dt. This calculation and a formal argument show that an LSI with a special potential function may be useful for the study of semi-classical behavior of low lying spec- trum of Schrodinger operators over a pinned path space. This kind of log-Sobolev inequality with special potential function already appeared in (13, 10). In Section 3, we consider a pinned path group and introduce an H-derivative on Pe,a(G) and the Dirichlet form in L 2 -space with respect to the (scaled) pinned Brownian motion measure with scaling (semi-classical) parameter ‚. The H-derivative is considered as the gradient operator on the path space which is defined by the right invariant