  At the lower edge of the energy continuum the birth of an isolated quantum
bound state is studied as caused by an infinitesimally small change of the
interaction. In our model a single, asymptotically free massive quantum
particle is assumed moving along a discretized real line of coordinates, $x \in
\mathbb{Z}$. The motion is assumed controlled by a weakly nonlocal
2J-parametric external potential which is non-Hermitian but PT-symmetric.
Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the
bound-state energy is treated as a variable real parameter while the value of
one of the couplings (responsible for the existence of the bound state) is
determined via the standard secular equation. It is found that in such an
arrangement the model is exactly solvable at all of the finite counts J of the
couplings. For illustration, the explicit closed bound-state formulae are
presented up to J=7.
