A new solvable complex PT-symmetric potential

• A non-analytic (no-differentiable) complex PT-symmetric potential.
• Real and complex-conjugate discrete energy eigenvalues.
• Entire discrete spectrum not real.
• Exceptional points.
• Flipping of eigenstates of complex-conjugate energy eigenvalues under PT.

We propose a new solvable one-dimensional complex PT-symmetric potential as V(x)=igsgn(x)|1−exp⁡(2|x|/a)| and study the spectrum of H=−d2/dx2+V(x)H=−d2/dx2+V(x). For smaller values of a,g<1a,g<1, there is a finite number of real discrete eigenvalues. As a and g   increase, there exist exceptional points (EPs), gngn (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 ψn(x)ψn(x) satisfy (1): PTψn(x)=1ψn(x) and (2): PTψEn(x)=ψEn⁎(x), for real and complex energy eigenvalues, respectively.