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

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. Then, Mourre’s main results are

• the point spectrum of ${H}$ in the interior of ${I}$ is finite.
• for any closed interval ${J \Subset I \cap \sigma_c(H)}$ and any ${z\in J}$, the operator ${\langle A \rangle^{-1} (H - z - i\epsilon)^{-1} \langle A\rangle^{-1}}$ is bounded on ${\mathcal{H}}$ uniformly as ${\epsilon \rightarrow 0}$, where ${\langle A \rangle = \sqrt{1+|A|^2}}$.

The Mourre’s theory is proven to be very useful in the study of spectral and scattering theory for Schrödinger operators and other dispersive PDEs. For instance, it yields the limiting absorbing principle, which in turn gives the Kato’s local smoothing estimate and the scattering RAGE’s theorem; for instance, see this blog post of T. Tao.

Below, I shall give a sketch of the proof of the Mourre’s beautiful theorem, then derive some local decay estimates on solutions to Schrödinger equations, and discuss some quick applications to linear damping in fluids.