Abstract: 
 First, we present optimal spectral enclosures for discrete Laplacians on $\mathbb{Z}$ and $\mathbb{N}$ with the Robin boundary condition perturbed by $\ell^{1}$-complex potentials. Second, we discuss results on a spectral stability of discrete Schrödinger operators on $\mathbb{N}$ with small complex potentials and related discrete Hardy inequalities. The talk is based on joint projects with O. O. Ibrogimov, D. Krejcirik, and A. Laptev.