# 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

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topShiffman, 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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