J.-P. Eckmann and M. Hairer
Spectral Properties of Hypoelliptic Operators
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ABSTRACT.  We study hypoelliptic operators with 
polynomially bounded coefficients that are of the form 
$K = \sum_{i=1}^m X_i^T X_i^{} + X_0 + f$, 
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 $\L^2$. For any $\eps>0$ we show that an inequality of the form $ 
\|u\|_{\delta,\delta} \le C\(\|u\|_{0,\eps} + \|(K+iy) u\|_{0,0}\)$ 
holds for suitable $\delta $ and $C$ which are independent of $y\in\R$, 
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 
H\'erau and Nier [HN02], we conclude that its spectrum lies in a 
cusp $\{x+iy~|~ x\ge |y|^\tau-c, \tau\in(0,1],c\in\R \}$.