Non-Self-Adjoint Quasi-periodic Operators with complex spectrum

Abstract

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