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

Abstract

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+4iVsin 2x that is a PT-symmetric optical potential, especially when |c|=|sqrt{1-4V^{2}}|<2 or correspondingly 0leq V<sqrt {5}/2. We obtain some useful equations for calculating Dirichlet eigenvalues also for |c|geq 2 or equally Vgeq sqrt{5}/2. We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger 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.