Note on quantum unique ergodicity

Zelditch, Steve

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

Note on quantum unique ergodicity
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by Steve Zelditch PDF

Proc. Amer. Math. Soc. 132 (2004), 1869-1872 Request permission

Abstract:

We prove that (near) off-diagonal matrix elements $\langle A \varphi _i, \varphi _j \rangle$ ($i \not = j$) of pseudodifferential operators relative to eigenfunctions of quantum unique- ly 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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