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Math. 340, 98–184. Amer. Math. org/mathph/0307062 89. : Quantum site percolation on amenable graphs. 317–328. org/math-ph/0308041 (2005) 90. : Spectral analysis of percolation Hamiltonians. Math. Ann. org/math-ph/0405006 91. : Linear Operators in Hilbert Spaces. Grad. Texts in Math. V. 2007 Abstract We prove local and global well-posedness for semi-relativistic, non√ linear Schrödinger equations i∂t u = − + m2 u + F(u) with initial data in H s (R3 ), s 1/2. Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions.

Recall from Section 5 that the measure ρ H on R is given by ρ H (ϕ) = τ (ϕ(H)) for continuous ϕ on R with compact support. 1 There exists an ⊂ of full measure and subsets of the real numbers and • , where • ∈ {disc, ess, ac, sc, pp, f in}, such that for all ω ∈ σ (Hω ) = and σ• (Hω ) = • Groupoids, von Neumann algebras and the IDS 33 for any • = disc, ess, ac, sc, pp, f in. Moreover, the almost-sure spectrum coincides with the topological support of ρ H . If is infinite, disc = ∅. If the group acting on X is amenable it was shown in [89] that property (P5), too, holds for the family (Hω )ω .

Math. 323, 53–67 (1981) 3. : Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–164 (1980) 4. : Almost periodic Schrödinger operators. I. Limit periodic potentials. Comm. Math. Phys. 82(1), 101–120 (1981/82) 5. : Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J. 50(1), 369–391 (1983) 6. : K-Theory of C∗ -algebras in solid state physics. , Winnink, M. ) Statistical Mechanics and Field Theory: Mathematical Aspects.