Distribution laws for integrable eigenfunctions
[Lois de distribution des fonctions propres intégrables]
Shiffman, Bernard ; Tate, Tatsuya ; Zelditch, Steve
Annales de l'Institut Fourier, Tome 54 (2004), p. 1497-1546 / Harvested from Numdam
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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.

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.

Publié le : 2004-01-01
DOI : https://doi.org/10.5802/aif.2057
Classification:  35P20,  32H30,  81S30 
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     author = {Shiffman, Bernard and Tate, Tatsuya and Zelditch, Steven},
     title = {Distribution laws for integrable eigenfunctions},
     journal = {Annales de l'Institut Fourier},
     volume = {54},
     year = {2004},
     pages = {1497-1546},
     doi = {10.5802/aif.2057},
     mrnumber = {2127856},
     zbl = {1081.35063},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_5_1497_0}
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Shiffman, Bernard; Tate, Tatsuya; Zelditch, Steve. Distribution laws for integrable eigenfunctions. Annales de l'Institut Fourier, Tome 54 (2004) pp. 1497-1546. doi : 10.5802/aif.2057. http://gdmltest.u-ga.fr/item/AIF_2004__54_5_1497_0/

[Be] M.V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977) p. 2083-2091 | MR 489542 | Zbl 0377.70014

[BHO] M.V. Berry, J. H. Hannay, A.M. Ozorio & De Almeida, Intensity moments of semiclassical wavefunctions, J. Phys. D 8 (1983) p. 229-242 | MR 724590

[De] T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France 116 (1988) p. 315-339 | Numdam | MR 984900 | Zbl 0676.58029

[FE] V. I. Falcko & K. B. Efetov, Statistics of wave functions in mesoscopic systems, J. Math. Phys 37 (1996) p. 4935-4967 | MR 1411615 | Zbl 0894.35092

[Fu] W. Fulton, Introduction to Toric Varieties, Annals of Math. 131, Princeton Univ. Press, 1993 | MR 1234037 | Zbl 0813.14039

[GKZ] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory and Applications, Birkhäuser, 1994 | MR 1264417 | Zbl 0827.14036

[Gu] V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T n -Spaces, Progress in Math. 122, Birkhäuser, 1994 | MR 1301331 | Zbl 0828.58001

[He] D. A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, IMA Math. Appl. Vol. 109, Springer-Verlag, 1999, p. 291-315 | Zbl 0982.11029

[HR] D. A. Hejhal & B. N. Rackner, On the topography of Maass waveforms for PSL(2,Z), Experiment. Math 1 (1992) p. 275-305 | MR 1257286 | Zbl 0813.11035

[Hö] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, Second Ed., Springer-Verlag, 1990 | Zbl 0712.35001

[Ka] N. M. Katz, Sato-Tate equidistribution of Kurlberg-Rudnick sums, Internat. Math. Res. Notices (2001) p. 711-728 | MR 1846353 | Zbl 1011.11058

[KR] P. Kurlberg & Z. Rudnick, Value distribution for eigenfunctions of desymmetrized quantum maps, Internat. Math. Res. Notices (2001) p. 985-1002 | MR 1860122 | Zbl 1001.81025

[LS] E. Lerman & N. Shirokova, Completely integrable torus actions on symplectic cones, Math. Res. Lett 9 (2002) p. 105-115 | MR 1892317 | Zbl 1001.37046

[Mi] A. D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, Phys. Rep 326 (2000) p. 259-382 | MR 1745139

[MF] A. D. Mirlin & Y. V. Fyodorov, Distribution of local densities of states, order parameter function, and critical behavior near the Anderson transition, Phys. Rev. Lett. 72 (1994) p. 526-529

[PA] V. N. Prigodin & B. L. Altshuler, Long-range spatial correlations of eigenfunctions in quantum disordered systems, Phys. Rev. Lett 80 (1998) p. 1944-1947

[STZ] B. Shiffman, T. Tate & S. Zelditch, Harmonic analysis on toric varieties, Contemporary Math 332, Amer. Math. Soc., p. 267-286 | Zbl 1041.32004

[SZ1] B. Shiffman & S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys 200 (1999) p. 661-683 | MR 1675133 | Zbl 0919.32020

[SZ2] B. Shiffman & S. Zelditch, Random polynomials with prescribed Newton polytope, J. Amer. Math. Soc 17 (2004) p. 49-108 | MR 2015330 | Zbl 02023441

[SS] M. Srednicki & F. Stiernelof, Gaussian fluctuations in chaotic eigenstates, J. Phys. A 29 (1996) p. 5817-5826 | MR 1419197 | Zbl 0905.58031

[TZ] J. Toth & S. Zelditch, L p -norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4 (2003) p. 343-368 | MR 1985776 | Zbl 1028.58028

[Y] S.-T. Yau, Open problems in geometry, Proc. Sympos. Pure Math. 54, Amer. Math. Soc., 1993, p. 1-28 | Zbl 0801.53001

[Ze] S. Zelditch, Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices (1998) p. 317-331 | MR 1616718 | Zbl 0922.58082