Communications in Mathematical Physics

, Volume 235, Issue 2, pp 233–253 | Cite as

Spectral Properties of Hypoelliptic Operators

  • J.-P. Eckmann
  • M. Hairer

Abstract:

 We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑i=1mXiTXi+X0+f, where the Xj denote first order differential operators, f is a function with at most polynomial growth, and XiT denotes the formal adjoint of Xi in L2. For any ɛ>0 we show that an inequality of the form ||u||δ,δC(||u||0,ɛ+||(K+iy)u||0,0) holds for suitable δ and C which are independent of yR, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Hérau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x≥|y|τc,τ(0,1],cR}.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • J.-P. Eckmann
    • 1
  • M. Hairer
    • 1
  1. 1.Département de Physique Théorique, Université de Genève, Geneve, Switzerland. E-mail: Jean-Pierre.Eckmann@physics.unige.ch; Martin.Hairer@physics.unige.chCH

Personalised recommendations