conjugate operator method

Mourre’s theory and local decay estimates, with some applications to linear damping in fluids

ttn12 fluid dynamics , , , ,

In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator {H} to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator {H} on a Hilbert space {\mathcal{H}} (e.g., {L^2} with the usual norm), and assume that there is a self-adjoint operator {A}, called a conjugate operator of {H} on an interval {I\subset \mathbb{R}}, so that

\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)

for some positive constant {\theta_I} and some compact operator {K} on {\mathcal{H}}, where {P_I} denotes the spectral projection of {H} onto {I}, the commutator {[H,A] = HA - AH}, and the inequality is understood in the sense of self-adjoint operators.

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