Distribution laws for integrable eigenfunctions

Bernard Shiffman[1]; Tatsuya Tate; Steve Zelditch

  • [1] Johns Hopkins University, Department of Mathematics, Baltimore, MD 21218 (USA), Department of Mathematics, Keio University, Keio University 3-14-1 Hiyoshi Kohoku-ku, Yokohama, 223-8522 (Japon)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1497-1546
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Shiffman, Bernard, Tate, Tatsuya, and Zelditch, Steve. "Distribution laws for integrable eigenfunctions." Annales de l’institut Fourier 54.5 (2004): 1497-1546. <http://eudml.org/doc/116150>.

@article{Shiffman2004,
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.},
affiliation = {Johns Hopkins University, Department of Mathematics, Baltimore, MD 21218 (USA), Department of Mathematics, Keio University, Keio University 3-14-1 Hiyoshi Kohoku-ku, Yokohama, 223-8522 (Japon)},
author = {Shiffman, Bernard, Tate, Tatsuya, Zelditch, Steve},
journal = {Annales de l’institut Fourier},
keywords = {toric Kähler variety; joint eigenfunction of the torus action; distribution law of the eigenfunction; semi-classical scaling limit; moments; line bundle; disordered system; pointwise asymptotics; Hecke eigenfunctions; phase space; configuration space},
language = {eng},
number = {5},
pages = {1497-1546},
publisher = {Association des Annales de l'Institut Fourier},
title = {Distribution laws for integrable eigenfunctions},
url = {http://eudml.org/doc/116150},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Shiffman, Bernard
AU - Tate, Tatsuya
AU - Zelditch, Steve
TI - Distribution laws for integrable eigenfunctions
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1497
EP - 1546
AB - 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.
LA - eng
KW - toric Kähler variety; joint eigenfunction of the torus action; distribution law of the eigenfunction; semi-classical scaling limit; moments; line bundle; disordered system; pointwise asymptotics; Hecke eigenfunctions; phase space; configuration space
UR - http://eudml.org/doc/116150
ER -

References

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  1. M.V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), 2083-2091 Zbl0377.70014MR489542
  2. M.V. Berry, J. H. Hannay, A.M. Ozorio, de Almeida, Intensity moments of semiclassical wavefunctions, J. Phys. D 8 (1983), 229-242 MR724590
  3. T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339 Zbl0676.58029MR984900
  4. V. I. Falcko, K. B. Efetov, Statistics of wave functions in mesoscopic systems, J. Math. Phys 37 (1996), 4935-4967 Zbl0894.35092MR1411615
  5. W. Fulton, Introduction to Toric Varieties, 131 (1993), Princeton Univ. Press, Princeton Zbl0813.14039MR1234037
  6. I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser, Boston Zbl0827.14036MR1264417
  7. V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T n -Spaces, 122 (1994), Birkhäuser, Boston Zbl0828.58001MR1301331
  8. D. A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, Emerging applications of number theory (Minneapolis, MN, 1996) Vol. 109 (1999), 291-315, Springer-Verlag, New York Zbl0982.11029
  9. D. A. Hejhal, B. N. Rackner, On the topography of Maass waveforms for P S L ( 2 , Z ) , Experiment. Math 1 (1992), 275-305 Zbl0813.11035MR1257286
  10. L. Hörmander, The Analysis of Linear Partial Differential Operators, I, (1990), Springer-Verlag, New York Zbl0712.35001
  11. N. M. Katz, Sato-Tate equidistribution of Kurlberg-Rudnick sums, Internat. Math. Res. Notices (2001), 711-728 Zbl1011.11058MR1846353
  12. P. Kurlberg, Z. Rudnick, Value distribution for eigenfunctions of desymmetrized quantum maps, Internat. Math. Res. Notices (2001), 985-1002 Zbl1001.81025MR1860122
  13. E. Lerman, N. Shirokova, Completely integrable torus actions on symplectic cones, Math. Res. Lett 9 (2002), 105-115 Zbl1001.37046MR1892317
  14. A. D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, Phys. Rep 326 (2000), 259-382 Zbl1013.81015MR1745139
  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), 526-529 
  16. V. N. Prigodin, B. L. Altshuler, Long-range spatial correlations of eigenfunctions in quantum disordered systems, Phys. Rev. Lett 80 (1998), 1944-1947 
  17. B. Shiffman, T. Tate, S. Zelditch, Harmonic analysis on toric varieties, Explorations in Complex and Riemannian Geometry 332, 267-286, Amer. Math. Soc., Providence, RI Zbl1041.32004
  18. B. Shiffman, S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys 200 (1999), 661-683 Zbl0919.32020MR1675133
  19. B. Shiffman, S. Zelditch, Random polynomials with prescribed Newton polytope, J. Amer. Math. Soc 17 (2004), 49-108 Zbl1119.60007MR2015330
  20. M. Srednicki, F. Stiernelof, Gaussian fluctuations in chaotic eigenstates, J. Phys. A 29 (1996), 5817-5826 Zbl0905.58031MR1419197
  21. J. Toth, S. Zelditch, L p -norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4 (2003), 343-368 Zbl1028.58028MR1985776
  22. S.-T. Yau, Open problems in geometry, Differential Geometry. Part 1: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990) 54 (1993), 1-28, Amer. Math. Soc., Providence, RI Zbl0801.53001
  23. S. Zelditch, Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), 317-331 Zbl0922.58082MR1616718

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