A new solvable complex PT-symmetric potential

We propose a new solvable one-dimensional complex PT-symmetric potential as V (x) = ig sgn(x) vertical bar 1 - exp(2 vertical bar x vertical bar/a)vertical bar 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), (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(n) (x) = Psi E-n* (x), for real and complex energy eigenvalues, respectively. (C) 2015 Published by Elsevier B.V.