Ergodicité et limite semi-classique. (Ergodicity and semi-classical limit). (French) Zbl 0624.58039

Let p(x,y) be a smooth function on phase space \({\mathbb{R}}^{2n}\) and denote by \(h>0\) Planck’s constant. The authors consider the corresponding quantum Hamiltonian \(P(h)=p(x,hD_ x)\) being selfadjoint on \(L^ 2({\mathbb{R}}^ n)\). If the classical flow on a compact energy shell \(\Sigma_{\lambda}=\{(x,y)|\) \(p(x,y)=\lambda \}\) is assumed to be ergodic and if some technical conditions are fulfilled then several interesting results are proved concerning the semiclassical limit of the quantum system. Especially the result is obtained that in the limit \(h\downarrow 0\) almost all (properly normalized) eigenfunctions \(\psi_ h\) of P(h) with eigenvalues \((=energies)\) near \(\lambda\) are distributed according to the Liouville measure \(\sigma_{\lambda}\) on the energy surface \(\Sigma_{\lambda}\), i.e. for any open set \(D\subset {\mathbb{R}}^ n\) it holds that \[ \lim_{h\downarrow 0} \int_{D} | \psi_ h(x)|^ 2 dx = \sigma_{\lambda}(\{(x,y)\in \Sigma_{\lambda}| x\in D\}). \] As a special case the one-dimensional Schrödinger equation is traced. Relations to WKB type results of other authors are discussed.
Reviewer: G.Jetschke


58Z05 Applications of global analysis to the sciences
37A99 Ergodic theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory
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