TY - JOUR

T1 - Small Scale Equidistribution of Random Eigenbases

AU - Han, Xiaolong

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity mλ of eigenfrequency λ tends to infinity at least logarithmically as λ → ∞. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of mλ. In particular, this implies that almost surely random eigenbases on the sphere Sn (n≥ 2) and the tori Tn (n≥ 5) are equidistributed at polynomial scales.

AB - We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity mλ of eigenfrequency λ tends to infinity at least logarithmically as λ → ∞. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of mλ. In particular, this implies that almost surely random eigenbases on the sphere Sn (n≥ 2) and the tori Tn (n≥ 5) are equidistributed at polynomial scales.

UR - http://www.scopus.com/inward/record.url?scp=84961218302&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2597-8

DO - 10.1007/s00220-016-2597-8

M3 - Article

SN - 0010-3616

VL - 349

SP - 425

EP - 440

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -