By G Venkataraman
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C. c. 2t and from the definition of the vector field we know that r − Id is in L∞ (X). s. ). 18) 2t Since r · Id − r is also in L∞ (X), the last term above gives also an O(t −1−δ ) contribution. c. + t + O(t −2+δ ) + O(t −1−δ ) + O(t −1−2δ ). s. t. t when taken in the mean value f . f t . Let us remark first that the product of the double commutator [iR, [iR, Qa ]] with some multiplication operators with functions of argument xt will appear quite frequently in this paper. We will prove below that if these functions are supported outside the subspaces Xb for all b ∈ L \ La then the desired decay in t of the terms containing such products will be ensured.
E−it Rα ψ by . s. 4) as follows: 1 t Q 2 g λt 1 gRα (P · ω + ω · P )Rα g λt 1 inf |x| g e−it Rα ψ 2 , x∈supp g t 1 Rα (P · ω + ω · P )Rα g e−it Rα ψ 2 . 4) we get 1 inf |x| − sup Rα x∈supp g λα 1 ∞ 2 −1,0 1 dt Q g t λt −it Rα e 2 ψ C h ∞ ψ 2. 5) x∈supp g applying the usual Fatou lemma (for the integral over t) yields: ∞ 1 Q −it R dt g e ψ t λt ∞ 2 = 1 Q −it Rα dt e ψ lim inf g α→∞ t λt ∞ lim inf α→∞ so the proposition is proved. 1 dt Q −it Rα g e ψ t λt 2 2 C ψ 2 2 Let us make some comments on this first a priori result.
7) holds. 3) holds also. Proof. Let us bring to mind first that given a vector operator S in H(X), we shall denote by Sa , S a the operators 1 ⊗a (πa S) , resp. (π a S) ⊗a 1 in H(X), but we shall not change the notation where the operators (πa S) acting in H(Xa ) and, respectively, (π a S) in H(Xa ) will be concerned. As stated before, whenever no confusion is possible, we shall not mention the usual time-dependent argument xt of the operators of multiplication with functions. 33 56 ANDREI IFTIMOVICI and compute as usual its Heisenberg derivative as: = DR 1 Re θ(R)(DR J ) iR, (Qa )2 J θ(R) + t (Qa )2 J θ(R).
A hot story by G Venkataraman