# Reviews in Mathematical Physics - Volume 12 by H. Araki, V. Bach, J. Yngvason (Editors)

By H. Araki, V. Bach, J. Yngvason (Editors)

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Now, for each n fixed, −T Pn (iσn (τ )) e−iξτ dτ is bounded in T if ξ is not of the form n · ωf , with n ∈ ZB . Hence, for ξ not of the form n · w f , n ∈ ZB , we have W (ξ) = 0. This completes the proof that exp(iF (t)) (and hence q(t)) is quasi-periodic with β(exp(iF (t))) = β(F ) = β(f ). ˜˜ ˜˜ ˜ ˜ Appendix C. The Relation Between q and f Since f is real and quasi-periodic we write f (t) = f0 + n∈ZB fn ein˜·ω˜f t , ˜ ˜n=0 ˜ ˜ with f0 = M (f ) ∈ R and fn = f−n . ˜ consider here the case where the sum above is To simplify our analysis˜ we will a finite sum.

30 J. C. A. 27) ˜(ωf , f0) ∈ RB+1 , if f˜0 = 0 , . ˜ ˜ ˜ , the definition above Since we are assuming that ωf ∈ RB says that all components + ˜ we will denote of ω are always non-zero. Moreover, ω := if f0 = 0 . 28) ˜ if f0 = 0 ˜ B B We will denote vectors in Z (or R ) by v and vectors in ZA (or RA ) by v. The symbol |n| will denote the l1 (ZA ) norm of a vector n = (n1 , . . , nA ) ∈ ZA : |n| := |n1 | + · · · + |nA |. We will use the symbol 1l for the identity matrix. Mat(n, C) is the set of all n × n matrices with complex entries.

Wreszinski and S. Casmeridis, “Models of two level atoms in quasiperiodic external fields”, J. Stat. Phys. 90 (1998) 1061–1068. [6] H. Jauslin and J. L. Lebowitz, “Spectral and stability aspects of quantum chaos”, Chaos 1 (1991) 114. [7] J. Feldman and E. Trubowitz, “Renormalization in classical mechanics and many body quantum field theory”, J. D’Analyse Math. 58 (1992) 213. 64 J. C. A. BARATA [8] L. H. Eliasson, “Absolutely convergent series expansions for quasi-periodic motions”, Math. Phys. Electronic J.