Mathematical Physics, Analysis and Geometry - Volume 10 by V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)

By V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)

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A) For ε > 0 and s ∈ R, we have that Mε is a bounded map from H s into H s+1 . (b) Mε u Hs u Hs whenever u ∈ H s and s ∈ R. s (c) For u ∈ H and s ∈ R, we have that Mε u → u strongly in H s as ε ↓ 0. We shall use tacitly properties (a)–(c) in the following analysis. By means of Mε and noting that E ∈ C1 (H 1/2 ; R), we can compute in a welldefined way for t1 , t2 ∈ [0, T) as follows t2 E[Mε u(t2 )]− E[Mε u(t1 )] = E (Mε u), Mε u˙ dt t1 t2 = t1 Re AMε u+ F(Mε u),−iMε (Au+ F(u)) dt Well-posedness for semi-relativistic Hartree equations of critical type t2 = t1 55 Im AMε u, Mε Au + F(Mε u), Mε Au + + AMε u, Mε F(u) + F(Mε u), Mε F(u) dt t2 =: t1 fε (t) dt, (25) where we write u = u(t) for brevity and recall the definition of F from (12).

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To show that an associated operator family ω → Hω has almost surely no discrete spectrum for infinite , we define the sequence of functions fn by fn (ω, γ ) := 1 χ I (γ ). |In | n Here, In is an exhaustion of the infinite group . The sequence ( fn ) satisfies property (21). 9 hold. 11 and conclude that the von Neumann algebra is of type I I. More precisely τ (Id) = E{tr(χD )} = p|D| < ∞ shows that the type is I I1 . Besides the discrete, essential, absolutely continuous, singular continuous, and pure point spectrum, σdisc , σess , σac , σsc , σ pp , the set σ f in consisting of eigenvalues which posses an eigenfunction with finite support is a quantity which may be associated with the whole family Hω , ω ∈ .

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