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We consider the wave and Schr\"odinger equations on a bounded open connectedsubset $\Omega$ of a Riemannian manifold, with Dirichlet, Neumann or Robinboundary conditions whenever its boundary is nonempty. We observe therestriction of the solutions to a measurable subset $\omega$ of $\Omega$ duringa time interval [0,T] with T>0. We address the problem of the optimal locationof the observation subset \omega among all possible subsets of a given measureor volume fraction. We solve it in two different situations. First, when aspecific choice of the initial data is given and therefore we deal with aparticular solution, we show that the problem always admits at least onesolution that can be regular or of fractal type depending on the regularity ofthe initial data. We then introduce a relevant criterion, not depending on theinitial conditions and to choose the observation set in an uniform way,independent of the data and solutions under consideration. Through spectraldecompositions, this leads to a second problem which consists of maximizing aspectral functional that can be viewed as a measure of eigenfunctionconcentration. This spectral criterion can be interpreted in two ways: itcorresponds either to a time asymptotic observability constant as theobservation time interval tends to infinity, or to a randomized version of thedeterministic observability inequality. Under appropriate quantum ergodicityassumptions on $\Omega$, we compute the optimal value. We then provide spectralapproximations and present some numerical simulations that fully confirm thetheoretical results in the paper and support our conjectures. Our resultshighlight precise connections between optimal observability issues and quantumergodic properties of the domain under consideration.

Optimal observability of the multi-dimensional wave and Schr\"odinger equations in quantum ergodic domains