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Let M be a complete Riemannian manifold with finite volume which we initially assume to be compact. Then since M is compact, L2(M) is spanned by the eigenfunctions of the Laplacian Δ on M . Many interesting questions can be asked about these eigenfunctions and their properties, and of these we focus on one, quantum ergodicity, which to the best of my knowledge was first considered by A.I. Šnirel′man, and was substantially sharpened in the work of Z. Rudnick and P. Sarnak which deals with the equidistribution properties of these eigenfunctions. Specifically, let φn be a complete orthonormal sequence of eigenfunctions of Δ ordered by eigenvalue. These can be interpreted for example as the steady states for Schroedinger’s equation

Adelic Dynamics and Arithmetic Quantum Unique Ergodicity