OAI.PlanarAnderson.anderson_spectrum_ae

The theorem states that for every real h>0, for almost every disorder configuration v under disorderLaw(h), there exists a bounded complex-linear operator H on the Hilbert space ℓ²(ℤ², ℂ) of square-summable complex functions on the planar lattice ℤ×ℤ such that H is an Anderson operator for v, H is self-adjoint, and the spectrum of H in ℂ equals the real interval [-4-h, 4+h] (embedded in ℂ as the points with imaginary part 0). Here a configuration is a real-valued function v on lattice sites, and disorderLaw(h) is the infinite product measure over sites of the uniform distribution (Lebesgue measure conditioned on the interval) on [-h,h], so the potential values are independent. H is an Anderson operator for v when, for every u in ℓ² and every site x=(a,b), (Hu)(a,b) = u(a+1,b)+u(a-1,b)+u(a,b+1)+u(a,b-1)+v(a,b)u(a,b), the sum of the four nearest-neighbour values plus the local potential times u at x. The statement is recorded as an admitted theorem.