We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d2/dx2 + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f ′′+μf = zf take center stage. We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures.

We consider in this work the initial value problem for the one dimensional cubic nonlinear Schrödinger equation. In order to integrate it numerically, one option frequently used, is to impose local absorbing boundary conditions. A finite element discretization in space of the cubic nonlinear Schrödinger equation is considered along with the absorbing boundary conditions obtained for an analogou...

Essentially, the Darboux proposition is based on the covariance properties of ordinary and partial differential equations with respect to a gauge transformation in the special case of second order differential equations of the SturmLiouville type. In this work, the one-dimensional Schrödinger equation with a position-dependent mass (SEPDM) is transformed into a Schrödinger-like equation with a ...

This thesis focuses on the development and the implementation of domain decomposition methods for the linear and non-linear, one dimensional and two dimensional Schrödinger equations. In the first part, we focus on the Schwarz waveform relaxation method (SWR) for the one dimensional Schrödinger equation. In the case the potential is linear and time-independent, we propose a new algorithm that i...

D−dimensional central and complex potentials of a Coulomb plus quartic-polynomial form are considered in a PT −symmetrized radial Schrödinger equation. Arbitrarily large finite multiplets of bound states are shown obtainable in an elementary form. Relations between their energies and couplings are determined by a finite-dimensional secular equation. The Bender’s and Boettcher’s one-dimensional ...

We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.