Non-self-adjoint quasi-periodic operators with complex spectrum

We give a precise and complete description on the spectrum for a class of non-self-adjoint quasi-periodic operators acting on ${\ell }^2\left(Z^d\right)$ which contains the Sarnak's model as a special case. As a consequence, one can see various interesting spectral phenomena including $PT$ symmetric breaking, the non-simply-connected two-dimensional spectrum in this class of operators. Particularly, we provide new examples of non-self-adjoint operator in ${\ell }^2\left(Z\right)$ whose spectra (actually a two-dimensional subset of $C$) can not be approximated by the spectra of its finite-interval truncations.