An estimate of the Gap of spectrum of Schrödinger operators which generate hyperbouded semigroups

Logarithmic Sobolev inequalities with potential functions on loop spaces were proved by L. Gross (1991, J. Funct. Anal. 102, 265-313), S. Aida (1996, in "Proceedings of the Fifth Gregynog Symposium, Stochastic Analysis and Applications," pp. 1-19, World Scientific, Singapore), and F.-Z. Gong and Z.-M. Ma (1998, J. Funct. Anal., 599-623). The generators of the Dirichlet forms are Schrödinger operators on loop spaces and they generate hyperbounded semigroups. Recently, F.-Z. Gong, M. Röckner and L. M. Wu (2000, Poincaré inequality for weighted first order Sobolev spaces on loop spaces, submitted) proved the existence of the gap of spectrum at the lowest eigenvalue of the Schrödinger operators by using M. Hino's exponential decay estimates of the semigroup (2000, Osaka J. Math. 37, 603-624) and the author's strong ergodicity property of the semi-group (S. Aida, 1998, J. Funct. Anal. 158, 152-185). In this paper, we will give a lower bound on the gap by using a weak Poincaré inequality which was introduced by M. Röckner and F.-Y. Wang (2000, Weak Poincaré inequalities and L²-convergence rates of Markov semigroups, preprint). Also we will give estimates on the distribution function of ground states using the weak Poincaré inequality.