A new solvable complex PT-symmetric potential

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. 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 $ψ_n(x)$ satisfy (1): PT$ψ_n(x)=1 ψ_n(x)$ and (2): PT$ψ_{E_n}(x)=ψ_{E^*_n}(x)$, for real and complex energy eigenvalues, respectively.

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