  We derive analytical results on energy spectral phase transitions and
deformations in the simplest model of one-dimensional lattice displaying the
non-Hermitian skin effect, namely the Hatano-Nelson model with unidirectional
hopping, under on-site potential uncorrelated disorder in complex energy plane.
While the energy spectrum under open boundary conditions (OBC) exactly
reproduces the distribution of on-site potential disorder, the energy spectrum
under periodic boundary conditions (PBC) undergoes spectral deformations, from
one or more closed loops in the fully delocalized phase, with no overlap with
the OBC spectrum, to a mixed spectrum (closed loops and some OBC energies) in
the mobility edge phase, to a complete collapse toward the OBC spectrum in the
bulk localized phase. Such transitions are observed as the strength of disorder
is increased. Depending on the kind of disorder, different interesting
behaviors are found. In particular, for continuous disorder with a radial
distribution in complex energy plane it is shown that in the delocalized phase
the energy spectrum under PBC is locked and fully insensitive to disorder,
while transition to the bulk localized phase is signaled by the change of a
topological winding number. When the disorder is described by a discrete
distribution, the bulk localization transition never occurs, while topological
phase transitions associated to PBC energy spectral splittings can be observed.
