Eigenvalue problems for the complex PT-symmetric potential V(x)=igxV(x)=igx

The spectrum of complex PT-symmetric potential, V(x)=igxV(x)=igx, is known to be null. We enclose this potential in a hard-box: V(|x|⩾1)=∞V(|x|⩾1)=∞ and in a soft-box: V(|x|⩾1)=0V(|x|⩾1)=0. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as En∼n2En∼n2. The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if |g|<(1.22330447)|g|<(1.22330447). The soft-box potential turns out to be a scattering potential not possessing reflectionless states.