Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential

We provide estimates for the eigenvalues of non-self-adjoint Sturm-Liouville operators with Dirichlet boundary conditions for a shift of the special potential 4cos(2)x+4iV sin2x that is a PT-symmetric optical potential, especially when |c|=| root 1-4V(2)<2 or correspondingly 0 <= V < root 5-/2. We obtain some useful equations for calculating Dirichlet eigenvalues also for |c|>= 2|or equally V >= 5-root/2. We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schr & ouml;dinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

Automated summary

This paper investigates the eigenvalues of non-self-adjoint Sturm-Liouville operators with Dirichlet boundary conditions for a PT-symmetric optical potential, particularly when |c|=| root 1-4V(2)<2 or correspondingly 0 <= V < root 5-/2. Estimates for the eigenvalues are provided, and the results are compared with the periodic and antiperiodic eigenvalues of the Schr & ouml;dinger operator. The complex eigenvalues are approximated using the roots of polynomials, and a numerical example with error analysis is presented.