UNIVERSAL ESTIMATE OF THE GAP FOR THE KAC OPERATOR IN THE CONVEX CASE

The aim of this paper is to prove that if V is a strictly convex potential with quadratic behavior at infinity, then the quotient mu2/mu1 between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L2(R(m)) by exp - V(x)/2.expDELTA(x).exp - V(x)/2, where DELTA(x) is the Laplacian on R(m) satisfies the condition: mu2/mu1 less-than-or-equal-to exp - cosh-1(sigma + 1)/2, where sigma is such that Hess V(x) greater-than-or-equal-to sigma > 0.