Операторы Шредингера с комплексным потенциалом на кубической решетке ( по совместной работе с А.Лаптевым и Я.Молером)

Schrodinger operators with complex potentials on the cubic lattice
We consider the Laplacian $\mathbb D$ on the lattice $\mathbb Z^d, d\ge 3$ and estimate the group $e^{it\mathbb D}$ and the resolvent $(\mathbb D-\mathbb l)^{-1}$ in the weighted spaces. The proof of the resolvent estimates is based on the investigation of the kernel of the resolvent. We obtain the estimate of the kernel of the resolvent $(\mathbb D-\mathbb l)^{-1}$ and their Hölder type estimates. We apply the obtained results to Schrödinger operators with real potentials and describe the scattering.
Moreover, we consider Schrödinger operators with complex decaying potentials on the lattice. We determine the trace formulas and estimate globally all zeros of the Fredholm determinant in terms of the potential.