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By Andre Martin, R. Schrader

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70)) we find (1 + ν)2 − (1 + σν)2 =: µ ˆ0 > 0 , µ ¯ 0 − µ0 ≥ (1 + ν)2 (1 + σν)2 (1 − σν)2 − (1 − ν)2 µ1 − µ ¯1 ≥ =: µ ˆ1 > 0 . 71) is bounded below by µ ≥ µ∗ = µ∗ (ν, α, σ) := min (1 − σ)ν , (1 − ν)2 ,µ ˆ0 , µ ˆ1 , (1 − σ)α > 0 . 75) holds with C∗ equal to k2 with k2 = k2 (ξ, ξ∗ , κ, τ, η0 , M, ν, α, σ) := C 2s 1 1 − ❡− ❡ m∗ (ν,α,σ) −1 . 26) hold. Let us turn to uniqueness. Let sˆ and C as in Proposition 3 and define k3 = k3 (ξ∗ , κ, τ, η0 , M, ν, α) := 2Cξ∗−ˆs , k4 := k2 k3 , and assume that k4 e ξ∗ ≤1.

103) Finally, we choose, as initial approximate solution, the trivial couple (v, β) := (0, 0) . 21) are given by V =1, W =1, ρ=0. 24), it suffices to require ε0 ≤ ε∗ := min 1 , 34 1 kM . 105) holds, by Theorem 3 and Remark 7, there exist unique functions u = u(θ; η) = uε (θ; η, ω) and γ = γ(η) = γε (η, ω) such that Fη (u; γ) = 0, for all η ∈ I0 , and θ → u(θ; η) ∈ H0ξ∗ . Furthermore, u and γ are Whitney C ∞ in all their variables (θ, η, ε, ω) in the domain T2ξ∗ × I0 × J × Dκ,τ , they are C ∞ in (θ, η, ε) and real–analytic in (θ; ε) ∈ T2ξ∗ × J.

106) where 0 < r < 1 is a prefixed number as in Theorem 2 and Ω−1 is the real–analytic function (inverse of e → Ωe ) defined in point (iii) of Remark 2. Clearly, 0 < e1 < e2 < 1 . 12)) are real–analytic function of (e, t) ∈ (0, 1) × S1 , where S1 := R/(2πZ). 107) such that the functions ρe (t) and fe (t) may be analytically continued into the complex domain Er,d × S1ξ¯, where Er,d := {e ∈ C : |e − e | ≤ d} , ¯ . 108) e ∈[e1 ,e2 ] 19 Again: do not confuse the letter e, which stands for eccentricity, with the letter e, which denotes the error function.

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Scattering theory: unitarity, analyticity and crossing by Andre Martin, R. Schrader


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