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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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References:

[1] M.V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A10 (1977) p. 2083-2091 · Zbl 0377.70014
[2] M.V. Berry, J. H. Hannay, A.M. Ozorio & de Almeida, Intensity moments of semiclassical wavefunctions, J. Phys. D8 (1983) p. 229-242
[3] T. Delzant, Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France116 (1988) p. 315-339 · Zbl 0676.58029
[4] V. I. Falcko & K. B. Efetov, Statistics of wave functions in mesoscopic systems, J. Math. Phys37 (1996) p. 4935-4967 · Zbl 0894.35092
[5] W. Fulton, Introduction to toric varieties, Annals of Math. 131, Princeton Univ. Press, 1993 · Zbl 0813.14039
[6] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory and Applications, Birkhäuser, 1994 · Zbl 0827.14036
[7] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian \(T^n\)-spaces, Progress in Math. 122, Birkhäuser, 1994 · Zbl 0828.58001
[8] 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
[9] D. A. Hejhal & B. N. Rackner, On the topography of Maass waveforms for \(PSL(2,Z),\) Experiment. Math1 (1992) p. 275-305 · Zbl 0813.11035
[10] L. Hörmander, The analysis of linear partial differential operators, I, Second Ed., Springer-Verlag, 1990 · Zbl 0712.35001
[11] N. M. Katz, Sato-Tate equidistribution of kurlberg-rudnick sums, Internat. Math. Res. Notices (2001) p. 711-728 · Zbl 1011.11058
[12] P. Kurlberg & Z. Rudnick, Value distribution for eigenfunctions of desymmetrized quantum maps, Internat. Math. Res. Notices (2001) p. 985-1002 · Zbl 1001.81025
[13] E. Lerman & N. Shirokova, Completely integrable torus actions on symplectic cones, Math. Res. Lett9 (2002) p. 105-115 · Zbl 1001.37046
[14] A. D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, Phys. Rep326 (2000) p. 259-382
[15] 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
[16] V. N. Prigodin & B. L. Altshuler, Long-range spatial correlations of eigenfunctions in quantum disordered systems, Phys. Rev. Lett80 (1998) p. 1944-1947
[17] B. Shiffman, T. Tate & S. Zelditch, Harmonic analysis on toric varieties, Contemporary Math 332, Amer. Math. Soc., p. 267-286 · Zbl 1041.32004
[18] B. Shiffman & S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys200 (1999) p. 661-683 · Zbl 0919.32020
[19] B. Shiffman & S. Zelditch, Random polynomials with prescribed Newton polytope, J. Amer. Math. Soc17 (2004) p. 49-108 · Zbl 1119.60007
[20] M. Srednicki & F. Stiernelof, Gaussian fluctuations in chaotic eigenstates, J. Phys. A29 (1996) p. 5817-5826 · Zbl 0905.58031
[21] J. Toth & S. Zelditch \(, L^p\)-norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré4 (2003) p. 343-368 · Zbl 1028.58028
[22] S.-T. Yau, Open problems in geometry, Proc. Sympos. Pure Math. 54, Amer. Math. Soc., 1993, p. 1-28 · Zbl 0801.53001
[23] S. Zelditch, Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices (1998) p. 317-331 · Zbl 0922.58082
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