Exact formulas of the end-to-end Greens functions in non-Hermitian systems

Greens function in non-Hermitian systems has recently been revealed to be capable of directional amplification in some cases. The exact formulas for end-to-end Greens functions are significantly important for both studies of non-Hermitian systems and their applications. In this work, based on the Widoms formula, we derive exact formulas for the end-to-end Greens functions which depend on the roots of a simple algebraic equation. These exact formulas allow direct and accurate comparisons between theoretical results and experimentally measured quantities. In addition, we verify the prior established integral formula in the bulk region to agree with the result in our framework.