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We derive analytical results on energy spectral phase transitions anddeformations in the simplest model of one-dimensional lattice displaying thenon-Hermitian skin effect, namely the Hatano-Nelson model with unidirectionalhopping, under on-site potential uncorrelated disorder in complex energy plane.While the energy spectrum under open boundary conditions (OBC) exactlyreproduces the distribution of on-site potential disorder, the energy spectrumunder periodic boundary conditions (PBC) undergoes spectral deformations, fromone or more closed loops in the fully delocalized phase, with no overlap withthe OBC spectrum, to a mixed spectrum (closed loops and some OBC energies) inthe mobility edge phase, to a complete collapse toward the OBC spectrum in thebulk localized phase. Such transitions are observed as the strength of disorderis increased. Depending on the kind of disorder, different interestingbehaviors are found. In particular, for continuous disorder with a radialdistribution in complex energy plane it is shown that in the delocalized phasethe energy spectrum under PBC is locked and fully insensitive to disorder,while transition to the bulk localized phase is signaled by the change of atopological winding number. When the disorder is described by a discretedistribution, the bulk localization transition never occurs, while topologicalphase transitions associated to PBC energy spectral splittings can be observed.

Spectral deformations in non-Hermitian lattices with disorder and skin effect: a solvable model