  The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk
topological invariants defined for infinite periodic systems and the occurrence
of protected zero-energy surface states in finite samples, is a ubiquitous and
widely observed phenomenon in topological matter. In non-Hermitian
generalizations of topological systems, however, this fundamental
correspondence has recently been found to be qualitatively altered, largely
owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary
conditions. In this work, we report on two contributions towards
comprehensively explaining this remarkable behavior unique to non-Hermitian
systems with theory. First, we analytically solve paradigmatic non-Hermitian
topological models for their zero-energy modes in the presence of generalized
boundary conditions interpolating between open and periodic boundary
conditions, thus explicitly following the breakdown of the conventional BBC.
Second, addressing the aforementioned spectral fragility of non-Hermitian
matrices, we investigate as to what extent the modified non-Hermitian BBC
represents a robust and generically observable phenomenon.
