Four-page commentary on the Herbst's contradictory Schr\"{o}dinger
equation which is ``exactly solvable" (in terms of Airy special
functions) but which only gives an empty spectrum of energies
under standard boundary conditions. In a way inspired by the needs
of the so called PT-symmetric Quantum Mechanics the author
contemplates two modifications of the model, viz., a toy version
(a) where the standard real line of coordinates shrinks to the
finite interval $(-1,1)$, and a toy version (b) where the
potential vanishes beyond  the finite interval $(-1,1)$. The
eigenvalue problem degenerates, in both cases, to the matching
condition tractable by Mathematica. In a way paralleling several
other similar studies (sampled here just by a self-quote: U.
G\"{u}nther et al, J. Math. Phys. 46 (2005) 063504), the author's
results on the model (a) are in full agreement with the
theoretical predictions and more general theorems  on the hard-box
PT-symmetric models proved by Langer and Tretter [Czechosl. J.
Phys. 54 (2004) 1113 and ibid. 56 (2006) 1063 (erratum)]. The
subsequent attention paid to model (b) reveals the absence of
reflectionless states and the existence of the two real
bound-state eigenvalues in a finite interval of couplings.

MR2288008
Ahmed, Zafar
Eigenvalue problems for the complex PT-symmetric potential $V(x)=igx$.
Phys. Lett. A 364 (2007), no. 1, 12--16. 81Q10 (81Q60)