Pseudospectral phenomena and the origin of the non-Hermitian skin effect

The non-Hermitian skin effect (NHSE), characterized by a macroscopic accumulation of eigenstates at the edge of a system with open boundaries, is often ascribed to a non-trivial point-gap topology of the Bloch Hamiltonian. We revisit this connection and show that the eigenspectrum of non-normal operators is highly sensitive to boundary conditions and generic perturbations, and therefore does not constitute a stable object encoding topological information. Instead, topological properties are reflected in the singular-value spectrum of finite systems and, in the semi-infinite limit, correspond to boundary-localized eigenmodes implied by the index of the corresponding Toeplitz operator. For a Hatano-Nelson ladder, where point-gap winding and non-normality can be varied independently, we demonstrate that the NHSE can occur without point-gap winding and, conversely, that point-gap winding can persist without the NHSE. These results establish that the NHSE originates from spectral instability and non-reciprocity rather than topology, and that the commonly assumed relation between spectral winding and boundary localization relies on translational invariance and is therefore not generic.