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- Title
Spectral Properties of Hypoelliptic Operators.
- Authors
Eckmann, J.-P.; Hairer, M.
- 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,δ≤C(||u||0,ℇ+||(K+iy)u||0,0) holds for suitable δ and C which are independent of y⋳R, 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|τ-c,τ⋳(0,1],c⋳R}.
- Subjects
HYPOELLIPTIC operators; PARTIAL differential operators; DIFFERENTIAL operators; DIFFERENTIAL equations; OPERATOR theory; FOKKER-Planck equation; PARTIAL differential equations; BESSEL functions
- Publication
Communications in Mathematical Physics, 2003, Vol 235, Issue 2, p233
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-003-0805-9