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**Additional info for Annales Henri Poincaré - Volume 11 **

**Example text**

44 M. S. Herman Ann. Henri Poincar´e From this it now follows that Δf, Δg ∈ D(eγ this, consider eγ x Δ f eγ x Δg 2 eγ eγ =4 eγ =4 eγ + x x x x x ), for any γ > 0. To see 2 [(V11 − E) f + V12 g] [V21 f + (V22 − E) g] (V11 − E) f + eγ V21 f + eγ x V12 g (V22 − E) g x 2 2 2 2 Let β > 0. Then, e2γ x 2 −2β |V21 f |2 dX dY ≤ V21 e x ∞ e2 So, eγ x V21 f ∈ L2 (R2 ) and by similar arguments eγ eγ x (V22 − E)g ∈ L2 (R2 ). Hence, eγ eγ x x Δf Δg x (γ+β) x f 2 2 <∞ (V11 − E)f, eγ x V12 g, 2 <∞ and Δf, Δg ∈ D(eγ x ).

Furthermore, since H2 (α) is unitarily equivalent to H2 for α ∈ R, we know that H2 (α) is self-adjoint and σ(H2 ) = σ(H2 (α)) for α ∈ R. 64 of [26]. 3 of section VII of [18], we know H2 (α) has compact resolvent for all α ∈ C. Thus it has purely discrete spectrum for all α ∈ C. Since H2 (α) is an entire analytic family in the sense of Kato, the eigenvalues are analytic on C except possibly at isolated crossings [26]. W (α) unitary implies that the eigenvalues are constant in a neighborhood of the real axis and thus crossings will not be an issue.

4, 5). While the splitting is nicely illustrated, we see that all of the series likely have radii of convergence well below 1. The radius of convergence appears to decrease as |l| or N increase. The divergent behavior was seen even at low orders of the perturbation coefﬁcients. We also used an elementary ﬁnite difference scheme to approximate the eigenvalues at several values of ˜b, for 0 < ˜b < 1. The results are given in Fig. 6. The plot was generated by approximating the lowest lying 17 eigenvalues for a ﬁxed ˜b value, then the value of ˜b was changed and the lowest 17 eigenvalues were calculated again.