Vincent Haugdahl: PT-symmetric Darboux Transformation

Abstract: In this thesis we study Darboux transformation of the zero potential Schrödinger equation. By considering complex-valued solutions and eigenvalues we obtain two distinct families of regular, complex-valued and PT-symmetric potentials not found in the literature. Formulas for the bound state eigenfunctions are provided and the associated scattering problems are solved. Both families of potentials are shown to decay exponentially fast and admit a finite number of bound states associated to the poles of the transmission coefficients, analogous to real-valued potentials in the Faddeev class.