## Distribution laws for integrable eigenfunctions.(English)Zbl 1081.35063

Authors’ abstract: We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kähler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.

### MSC:

 35P20 Asymptotic distributions of eigenvalues in context of PDEs 32H30 Value distribution theory in higher dimensions 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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