Quantum Ergodic Sequences and Equilibrium Measures

Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We generalize the definition of a “quantum ergodic sequence” of sections of ample line bundles L→ M from the case of positively curved Hermitian metrics h on L to general smooth metrics. A choice of smooth Hermitian metric h on L and a Bernstein–Markov measure ν on M induces an inner product on H0(M, LN). When ||sN||L2=1, quantum ergodicity is the condition that |sN(z)|2dν→dμφeq weakly, where dμφeq is the equilibrium measure associated with (h, ν). The main results are that normalized logarithms 1Nlog|sN|2 of quantum ergodic sections tend to the equilibrium potential, and that random orthonormal bases of H0(M, LN) are quantum ergodic.

Original languageEnglish (US)
Pages (from-to)89-118
Number of pages30
JournalConstructive Approximation
Volume47
Issue number1
DOIs
StatePublished - Feb 1 2018

Keywords

  • Bergman/Szegő kernel
  • Equilibrium measure
  • Line bundle
  • Quantum ergodic sequence
  • Random holomorphic section

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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