Distribution laws for integrable eigenfunctions
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1497-1546.

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.

On détermine l’asymptotique semi-classique des fonctions propres jointes de l’action d’un tore sur une variété kählérienne torique. Ces variétés sont des modèles de systèmes complètement intégrables en géométrie complexe. On démontre que les fonctions propres ressemblent ponctuellement à des Gaussiennes centrées aux tores correspondants. De plus, on prouve qu’il existe une limite universelle Gaussienne de la fonction de distribution renormalisée auprès de son centre, et on détermine sa distribution limite non-universelle loin de son centre.

DOI: https://doi.org/10.5802/aif.2057
Classification: 35P20,  32H30,  81S30
Keywords: toric Kähler variety, joint eigenfunction of the torus action, distribution law of the eigenfunction, semi-classical scaling limit
@article{AIF_2004__54_5_1497_0,
     author = {Shiffman, Bernard and Tate, Tatsuya and Zelditch, Steve},
     title = {Distribution laws for integrable eigenfunctions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1497-1546},
     doi = {10.5802/aif.2057},
     zbl = {1081.35063},
     mrnumber = {2127856},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2004__54_5_1497_0/}
}
Shiffman, Bernard; Tate, Tatsuya; Zelditch, Steve. Distribution laws for integrable eigenfunctions. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1497-1546. doi : 10.5802/aif.2057. https://aif.centre-mersenne.org/item/AIF_2004__54_5_1497_0/

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