Quantum unique ergodicity

Details Quantum unique ergodicity Quantum unique ergodicity

2003-01-27

arxiv.org/abs/math-ph/0301035v2

Proc. Amer. Math. Soc. 132 (2004), no. 6, 1869--1872

Physics

Mathematical Physics

Findal corrections for publication in Proc. AMS

Scientific paper

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an o(1) error). Bouncing ball quasimodes of the stadium are believed to have this property (E.J. Heller et al) and so are analogous quasimodes recently constructed by H. Donnelly on certain non-positively curved surfaces. The main ingredient is the proof that all sequences of off-diagonal matrix elements of QUE systems with vanishing spectral gaps tend to zero.

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