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We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators |vn〉〈vn| multiplied byg tan π[(α,n)+ω],g>0, α∈ℝd,n∈ℤd, ωε[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsvn(x)=v(x−n),d∈ℝd,n∈ℤd, is small then the spectrum ofH consists of a dense point component coinciding with [ϱ,∞) and an absolutely continuous component coinciding with [ϱ, ∞), where ϱ is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=ϱ) and zero temperature a.c. conductivity ϱ, 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.

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