TY - JOUR
T1 - Distribution of zeros of random and quantum chaotic sections of positive line bundles
AU - Shiffman, Bernard
AU - Zelditch, Steve
PY - 1999
Y1 - 1999
N2 - We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns "random" sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {SNj} of H0(M, LN), we show that for almost every sequence {SNj}, the associated sequence of zero currents 1/NZSNj tends to the curvature form ω of L. Thus, the zeros of a sequence of sections SN ∈ H0(M, LN) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {SNj} of H0(M, LN) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.
AB - We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns "random" sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {SNj} of H0(M, LN), we show that for almost every sequence {SNj}, the associated sequence of zero currents 1/NZSNj tends to the curvature form ω of L. Thus, the zeros of a sequence of sections SN ∈ H0(M, LN) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {SNj} of H0(M, LN) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.
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U2 - 10.1007/s002200050544
DO - 10.1007/s002200050544
M3 - Article
AN - SCOPUS:0033514098
VL - 200
SP - 661
EP - 683
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 3
ER -