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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=1 m X i T X i +X0+f, where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i T denotes the formal adjoint of X i 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}.

Keywords

Differential Operator Sobolev Space Spectral Property Heat Bath Polynomial Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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