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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Extra resources for Reviews in Mathematical Physics - Volume 12
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.  H. Jauslin and J. L. Lebowitz, “Spectral and stability aspects of quantum chaos”, Chaos 1 (1991) 114.  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  L. H. Eliasson, “Absolutely convergent series expansions for quasi-periodic motions”, Math. Phys. Electronic J.