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Quantum Ergodic Sequences and Equilibrium Measures

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Abstract

We generalize the definition of a “quantum ergodic sequence” of sections of ample line bundles \(L \rightarrow M\) 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 \(\nu \) on M induces an inner product on \(H^0(M, L^N)\). When \(||s_N||_{L^2} =1\), quantum ergodicity is the condition that \(|s_N(z)|^2 d\nu \rightarrow d\mu _{\varphi _{eq}} \) weakly, where \(d\mu _{\varphi _{eq}} \) is the equilibrium measure associated with \((h, \nu )\). The main results are that normalized logarithms \(\frac{1}{N} \log |s_N|^2\) of quantum ergodic sections tend to the equilibrium potential, and that random orthonormal bases of \(H^0(M, L^N)\) are quantum ergodic.

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Notes

  1. Also written \(\omega _{\varphi } \) with \(\varphi = - \log h\).

  2. Both notations \(\varphi _{eq} \) and \( V^*_{h, K}\), and also \(P_K(\varphi )\), are standard and we use them interchangeably. \(V^*_{h,K}\) is called the pluri-complex Green’s function in [10] and elsewhere.

  3. In other words, is \(U_N^* U_N = \Pi _N + o(1)\), where o(1) is measured in the operator norm?

  4. A pluripolar set is a subset of the \(-\infty \) set of a plurisubharmonic function.

  5. The notation \(B_{h^N, \nu }(z,z)\) is used in articles of Berman; \(\Pi _{h^N, \nu }(z)\) is the contraction of the diagonal \(\Pi _{h^N, \nu }(z,z)\).

  6. \(F_N\) is denoted \(\mathcal {L}_N\) in [8].

  7. Thanks to Turgay Bayraktar for the reference and explanations of this point.

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Acknowledgements

Thanks to R. Berman for comments on an earlier version, in particular for emphasizing that Definition 1.1 should be consistent with the expected mass formula for random sequences. Thanks also to T. Bayraktar for remarks and references on Theorem 3.5, and to the referees for many comments that helped improve the exposition.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Steve Zelditch

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  1. Steve Zelditch
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Correspondence to Steve Zelditch.

Additional information

Communicated by Sylvia Serfaty.

Research partially supported by NSF Grant and DMS-1541126 and by the Stefan Bergman trust.

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Zelditch, S. Quantum Ergodic Sequences and Equilibrium Measures. Constr Approx 47, 89–118 (2018). https://doi.org/10.1007/s00365-017-9397-z

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  • DOI: https://doi.org/10.1007/s00365-017-9397-z

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