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Extra resources for Communications in Mathematical Physics - Volume 249
The point spectrum of H is locally finite (counting multiplicity). Limiting absorption principle. 5. Assume Hypotheses (H0), (I1) and (I2). Let I ⊂ R\σpp (H ) be a compact interval. Then for 21 < s ≤ 1 the limits: 1 1 (N + 1) 2 (d (b) + 1)−s R(λ ± i0)(d (b) + 1)−s (N + 1) 2 1 1 := lim (N + 1) 2 (d (b) + 1)−s (H − λ ∓ iµ)−1 (d (b) + 1)−s (N + 1) 2 µ→0+ exist in norm uniformly in λ ∈ I . In particular σsc (H ) = ∅. Moreover, the maps: I 1 1 λ → (N + 1) 2 (d (b) + 1)−s R(λ ± i0)(d (b) + 1)−s (N + 1) 2 ∈ B(H) are H¨older continuous of order s − 1 2 for the norm topology of B(H).
Let h be a Hilbert space, which we will call the one-particle space. Let n (h) := ⊗ns h be the symmetric nth tensor power of h. Let Sn be the orthogonal projection of ⊗n h onto n (h). The Fock space over h is the direct Hilbert sum (h) := ∞ and n=0 n (h). The vacuum vector (1, 0 . . ) ∈ (h) will be denoted by the number operator N is defined as N |⊗ns h = nI. g. [DG1, Sect. 2]). n If g ⊂ h is a vector space, we denote by fin (g) ⊂ (h) the space ⊕∞ 0 ⊗s g, where direct sums and tensor products are taken in the algebraic sense.
Then for all ∗ f ∈ D(a (v)) one has 1 |(f, a ∗ (v)f )| ≤ C1 (r, v) (K + r) 2 ⊗ I (h) f 1 IK ⊗ d (ω) 2 f . 4. Additional remarks on the spaces B(L, K ⊗ h) and B(K, L∗ ⊗ h). We first give an alternative description of the spaces B(L1 , L2 ⊗ h) in the important particular case where h = L2 (Rd , dk) and L1 , L2 are separable Hilbert spaces. Let us denote by L2w (Rd ; B(L1 , L2 )) the space of (equivalence classes of) strongly measurable maps v(·) : Rd → B(L1 , L2 ) such that the function k → v(k)ψ 2 is integrable for all ψ ∈ L1 , and let us equip it with the norm v(·) = sup ψ =1 Rd v(k)ψ 2 dk 1 2 .