Note on quantum unique ergodicity

Steve Zelditch

doi:10.1090/S0002-9939-03-07298-8

Note on quantum unique ergodicity

Steve Zelditch^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We prove that (near) off-diagonal matrix elements script A sign _ψi, ψ _j (i ≠ j) of pseudodifferential operators relative to eigenfunctions of quantum uniquely ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.

Original language	English (US)
Pages (from-to)	1869-1872
Number of pages	4
Journal	Proceedings of the American Mathematical Society
Volume	132
Issue number	6
DOIs	https://doi.org/10.1090/S0002-9939-03-07298-8
State	Published - Jun 2004

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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Cite this

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title = "Note on quantum unique ergodicity",

abstract = "We prove that (near) off-diagonal matrix elements script A sign ψi, ψ j (i ≠ j) of pseudodifferential operators relative to eigenfunctions of quantum uniquely ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.",

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AB - We prove that (near) off-diagonal matrix elements script A sign ψi, ψ j (i ≠ j) of pseudodifferential operators relative to eigenfunctions of quantum uniquely ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.

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Note on quantum unique ergodicity

Abstract

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