300 énigmes by Nicolas Conti

By Nicolas Conti

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1) with Ωt,μ replacing Ωt . 109) op T0 ≤ μ + 64e6r0 T0 B0 Wτ (μ) − Wτ 2 2 0 op dτ . 6). 111) Wt (μ) − Wt sup t∈[0,T0 ] ≤ op 1 − e−2T0 μ +2T0 sup 1 + 2T02 B0 Ωt,μ − Ωt t∈[0,T0 ] +2r0 T0 2 2 op Wt (μ) − Wt sup op t∈[0,T0 ] . 112) 1 + 2T02 B0 Ωt,μ − Ωt t∈[0,T0 ] op 2 2 . 114) sup Ωt,μ − Ωt t∈[0,T0 ] op ≤ 2μ + 4 1 − e−2T0 μ B0 2 . 116) ∀t ∈ [0, T0 ] : (ϕ, Ωt ϕ) = lim+ (ϕ, Ωt,μ ϕ) ≥ 0 , μ→0 because of Lemma 38 applied to the operator family (Ωt,μ )t∈[0,T0 ] . In other words, the operator Ωt is positive for any t ∈ [0, T0 ].

155) lim+ (Wt,s − 1)Bs t→s 2 2 = lim ht,s (n) n=1 , t→s+ provided ht,s (n) has a limit for all n ∈ N, as t → s+ . By Lemma 33 (ii), the operator Wt,s is strongly continuous in t with Ws,s = 1. 157) lim (Wt,s − 1) Bs t→s+ 2 2 =0. 159) t − 1) Wt,s Bs (Wt,s 2 2 op t ∗ ¯s ≤ ((Wt,s ) − 1)B Licensed to Tulane Univ. 78. org/publications/ebooks/terms ≤ 1 and the cyclicity 2 2 46 V. TECHNICAL PROOFS ON THE ONE–PARTICLE HILBERT SPACE ∗ ∗ t t and Wt,s is also strongly continuous in t ≥ s ≥ 0 with Ws,s = 1 and t ≤ 1.

9). √ Proof. 19) take ω−1 := 1, ω1 := 2 and b1 ∈ (1/2, 1/ 2), while for k ∈ N\{1}, choose bk := 1/2k , ω−k := ωk := (3/4)k . Similar to the proof of Proposition 2 28, Conditions A1–A3 and Ω−2 0 B0 ∈ L (h) are clearly satisfied for this choice. 22). 33) requires for k = 1 that 1 − b21 > 0 and 1 − 4b21 > 0. 33) does not hold. 6) for k = 1 and k ∈ N\{1} separately. 38) (1) Ω0 = ω−1 0 0 ω1 and (1) B0 = (1) 0 b1 b1 0 . 24) for k = 1 and μ < 1− (2) (2) 2b21 , where b21 < 1/2. 26), that satisfy A1–A3 and A5.

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