Schroedinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist

We consider the Schroedinger-like operator H in which the role of a potential is played by the lattice sum of rank 1 operators vertical strokevn>0, αelement ofRd, nelement ofZd, ωelement of[=, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functions vn(x)=v(x-n), xelement ofRd, nelement ofZd, small then the spectrum of H consists of a dense point component coinciding with R and an absolutely continuous component coinciding with [ρ, ∞), where ρ is the radius of the mentioned support. Besides, we find the integrated density of states N(λ) (it has a jump at λ=ρ) and zero temperature a.c. conductivity σλ(ν), that also has a jump at λ=ρ and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠ρ. (orig.)