On the spectrum of the generator of an infinite system of interacting diffusions

We study the spectrum of the operator Lf(Q) = - ∑x∈ℤ^d (∂²f/∂q²_x) (Q) - β ∑x∈ℤ^d (∂H/∂q_x) (Q) (∂f/∂q_x) (Q), Q = {q_x}, generating an infinite-dimensional diffusion process Ξ(t), in space L₂ (ℝ^ℤd, dv(Q)). Here v is a "natural" Ξ(t)-invariant measure on ℝ^ℤd which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace H₁ ⊂ L₂(ℝ^ℤd, dv(Q)) such that L | H₁ has a distinctive character related to a "quasi-particle" picture. In particular, L | H₁ has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K₁ > 0 giving the smallest non-zero eigenvalue of a limiting problem associated with β = 0. An immediate corollary of our result is an exponentially fast L₂-convergence to equilibrium for the process Ξ(t) for small values of β.