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Consider the two notions of quantum ergodicity of the Laplacian operator $\Delta$.

(Phase space): $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there exists a density one index set of the natural numbers such that $$\lim_{k\to\infty}\left<a(x, E_{j_k}^{-1/2}D)u_{j_k}, u_{j_k}\right> = \frac{1}{|S^*X|}\int_{S^*X}a(x, \xi)d\mu$$

for all semiclassical pseudodifferential operators properly supported in the interior of $X$, where $\left(u_j\right)_{j \in B}$ is a sequence of orthonormal eigenfunctions of $\Delta$ and $S^*X$ is the cosphere bundle of $X$. As a side note, nearly all text I have seen in semiclassical analysis which discusses ergodicity revolves specifically around quantizised observables and hence quantum ergodicity in the phase space.

(Physical space:) $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there exists a density one index set of the natural numbers such that the sequence of probability measures formed from the eigenfunctions , i.e. $|u_{j_k}|^2$, converge to the uniform norm in weak-* topology, i.e. $\int_X f|u_{j_k}|^2dx = \int_X fdx$.

(Question:) What if any is the connection between the two notions of quantum ergodicity in the sense that if you know something about, say, the phase space version, so you also know something about the "physical space" (or weak-*) version?

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