### Abstract

We study the spectrum of the operator Lf(Q) = - ∑x∈ℤ^{d} (∂^{2}f/∂q^{2}_{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 L_{2} (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 H_{1} ⊂ L_{2}(ℝ^{ℤd}, dv(Q)) such that L | H_{1} has a distinctive character related to a "quasi-particle" picture. In particular, L | H_{1} has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K_{1} > 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 L_{2}-convergence to equilibrium for the process Ξ(t) for small values of β.

Original language | English (US) |
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Pages (from-to) | 463-489 |

Number of pages | 27 |

Journal | Communications In Mathematical Physics |

Volume | 206 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1999 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics