A new solvable complex PT-symmetric potential

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)=igsgn(x)\mathrm{\mid }1-\exp (2\mathrm{\mid }x\mathrm{\mid }\mathrm{/}a)\mathrm{\mid }$ and study the spectrum of $H=-d^2\mathrm{/}d{x}^2+V(x)$. For smaller values of $a,g<1$, there is a finite number of real discrete eigenvalues. As $a$ and $g$ increase, there exist exceptional points (EPs), $g_n$ (for fixed values of $a$) causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates ${\psi }_n(x)$ satisfy (1): PT${\psi }_n(x)=1{\psi }_n(x)$ and (2): PT${\psi }_{E_n}(x)={\psi }_{E_n^{\ast }}(x)$, for real and complex energy eigenvalues, respectively.