Abstract. We study the perturbation theory for H=pZ+x2+flx 2~+1, n= 1,2 ..... It is proved that when Imfl~:0, H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schr6dinger perturbation expansion, and admits an analytic continuation to Im/~=0 where it can be interpreted as a resonance of the problem.
We present a new proof of the convergence of the N −particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in , up to an exponentially small remainder. For = 0, the classical dynamics in the mean-field limit is given by the Vlasov equation.
Abstract. We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ 2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H0 + ǫP (ωt) for ǫ small. Here H0 is the one-dimensional Schrödinger operator p 2 + V , V (x) ∼ |x| α , α > 2 for |x| → ∞, the time quasi-periodic perturbation P may grow as |x| β , β < (α − 2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non constant coefficients.
The canonical quantization of any hyperbolic symplectomorphism A of the 2-torus yields a periodic unitary operator on a JV-dimensional Hubert space, N =•£. We prove that this quantum system becomes ergodic and mixing at the classical limit (N -• oo, N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly spread in phase space. Contents 1. Von Neumann definition of the quantum ergodicity and mixing properties. Statement of the main results 473 2. Koopman operator on invariant lattices and periodic orbits. Limits of atomic invariant measures supported on periodic orbits via Kloosterman sums 477 3. Quantization of toral automorphisms. Discrete Wigner functions. Support on classical periodic orbits, relation with the Koopman operator and explicit construction of the quantum eigenvectors 482 4. Classical limit of the matrix elements of the observables via Weil-Deligne exponential sums. Weak-* convergence of the Wigner functions. Proof of the main results 495 Appendix A. Some basic results out of number theory 503
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