Trace formulae for Schr\"odinger operators with complex-valued potentials on cubic lattices

TL;DR

This paper derives trace formulae for Schr"odinger operators with complex potentials on cubic lattices, enabling the estimation of all zeros of the Fredholm determinant based on the potential.

Contribution

It introduces new trace formulae for complex-valued potentials on lattices and provides methods to estimate zeros of the Fredholm determinant.

Findings

Derived trace formulae for complex potentials

Established estimates for zeros of the Fredholm determinant

Connected spectral properties to potential decay

Abstract

We consider Schr\"odinger operators with complex decaying potentials on the lattice. Using some classical results from Complex Analysis we obtain some trace formulae and using them estimate globally all zeros of the Fredholm determinant in terms of the potential.