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 -

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