Schrödinger operator with a complex steplike potential

The purpose of this article is to study pseudospectral properties of the one-dimensional Schrödinger operator perturbed by a complex steplike potential. By constructing the resolvent kernel, we show that the pseudospectrum of this operator is trivial if and only if the imaginary part of the potential is constant. As a by-product, a new method to obtain a sharp resolvent estimate is developed, answering a concern of Henry and Krejčiřík, and a way to construct an optimal pseudomode is discovered, answering a concern of Krejčiřík and Siegl. This article also analyzes the impact of a complex point interaction on the spectrum and the resolvent norm.