Spectral Properties of Hypoelliptic Operators

TL;DR

This paper investigates the spectral properties of hypoelliptic operators with polynomially bounded coefficients, establishing resolvent estimates and applying them to analyze the spectrum of the Fokker-Planck operator in a physical model.

Contribution

It provides new resolvent estimates for a class of hypoelliptic operators and characterizes the spectrum of the Fokker-Planck operator for an anharmonic oscillator chain.

Findings

Established resolvent inequalities in weighted Sobolev spaces.

Proved the spectrum of the Fokker-Planck operator lies in a specific cusp-shaped region.

Extended spectral analysis to models with polynomially bounded coefficients.

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 e > 0 we show that an inequality of the form |u|_{delta,delta} <= 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 Herau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x >= |y|^tau-c, tau in (0,1], c in R}.