A new solvable complex PT-symmetric potential

Abstract

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). 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), gn (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) satisfy (1): PTψn(x)=1ψn(x) and (2): PTψEn(x)=ψEn⁎(x), for real and complex energy eigenvalues, respectively.