Abstract
Consider a self adjoint quantic hamiltonian:P(h)=p(x, hD x) whereh>0 is the Planck's constant andp some smooth classical observable on the phase space R2n. When the classical flow on a compact energy shell {p=λ} is ergodic we prove that in the limith ↓ 0 almost all the eigenfunctions ofP(h) whose energy is near of λ are distributed according to the Liouville measure on {p=λ}.
In the high energy case (λ →+∞) this sort of problem was considered by A. Schnirelman, S. Zelditch, and Y. Colin de Verdière.
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Communicated by B. Simon
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Helffer, B., Martinez, A. & Robert, D. Ergodicité et limite semi-classique. Commun.Math. Phys. 109, 313–326 (1987). https://doi.org/10.1007/BF01215225
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DOI: https://doi.org/10.1007/BF01215225