We computed nonperturbatively, via the state-specific matrixcomplex eigenvalue Schrödinger equation (CESE) theory, theenergy shifts and widths of all ten Li n = 4 levels which areproduced by electric fields of strength, F, in the range 0.0-0.0012 au (6.17×106 V cm-1). Byestablishing the nature of the state vector to which everyphysically relevant complex eigenvalue corresponds, wedelineated the field strength region where it is possible tocharacterize the perturbed levels in terms of smallsuperpositions of unperturbed Li states (called here the weakfield region) from the region where the mixing of the discreteand the continuous states renders such an identificationimpossible (the strong field region). For both regions, systematic and accurate CESE calculations have produced acomplete adiabatic spectrum of perturbed energies. It turns outthat the behaviour of the energies of the m = ±1, ±2and ±3 levels is smooth. In contrast, the Starkspectrum of the m = 0 levels contains not only all the n = 4levels but also parts of the n = 3 and 5 manifolds, andis characterized by regions of crossing as well as of avoided crossing of the real part of the complex energies as a functionof F.In the former case, the widths of the crossing (diabatic)levels differ considerably - even by two orders of magnitude. Inthe latter case, there are abrupt changes in the widths, aresult of significant changes in the perturbed wavefunctions.For weak fields, which, for example, for the m = 0, n = 4levels correspond to values up to about 2×10-4 au, theone-electron semiclassical Ammosov, Delone and Krainov (ADK)formula for the widths produces reasonable trends. However, when the wavefunction mixingincreases with field strength, it fails completely.For the four m = 0 levels, comparison is made with the results of Sahoo and Ho, obtained nonperturbatively byapplying the complex absorbing potential (CAP) method. For arange of relatively weak fields, the CAP results for the widthsdo not follow a physically meaningful curve and deviate fromour CESE results by orders of magnitude. As regards the energyspectrum, the one given by Sahoo and Ho for strengths up to0.0005 au and the one given by us are substantially different, theformer presenting a simple picture, without any avoided crossings. Given these differences, we take the opportunity to comment, via exemplars, on problems and solutions pertaining tothe calculation of resonance states of polyelectronic atoms andmolecules in terms of non-Hermitian approaches employingsquare-integrable function spaces.