The linear PT-symmetric complex potential

Abstract

The spectrum of a PT-symmetric complex-valued linear potential is investigated. Working in the D =1 dimensional space, we consider V(x)= jxj+icx. Semianalytical solutions are given by using the properties of the Airy functions. The numerical integration of the differential equation system is discussed. We show that the number of eigenstates with a real eigenvalue is limited, depending on the ratio c= and on the quantum number n. This is reflecting the spontaneous breaking of PT symmetry. For the ground state, we conjecture the eigenvalue to be real for any value of c