Solvable PT-symmetric model with a tunable interspersion of nonmerging levels

We study the spectrum in such a PT-symmetric square well (of a diameter L⩽∞) where the “strength of the non-Hermiticity” is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance ℓ<L of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a nonempty interval of g⩽g0(ℓ,L). Surprisingly, a specific distinction between the bound states is found in their asymptotic stability∕instability with respect to an unlimited growth of g beyond g0(ℓ,L). In our model, all of the low-lying levels remain asymptotically unstable at the small ℓ⪡L and finite L while only the stable levels survive near ℓ≈L<∞ or in the purely imaginary force limit with 0<ℓ<L=∞. In between these two extremes, an unusual and tunable, variable pattern of the interspersed “robust” and “fragile” subspectra of the real levels is obtained.