with the Riesz space-fractional derivative of order 0 < α ≤ 2 in the presence of the linear potential V(x) = βx. The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1, which are useable f...

A numerical scheme for the one dimensional stationary Schrödinger-Poisson model is described. The scheme is used to simulate a Resonant Tunneling Diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discr...

It is shown that, for a large class of statistical mixtures, the Wigner-Poisson (or Hartree) system can be reduced to an effective Schrödinger-Poisson system, in which the Schrödinger equation contains a new nonlinearity. For the case of a zero-temperature one-dimensional electron gas, this additional nonlinearity is of the form Ψ 4. In the long wavelength limit, the results obtained from the e...

An adaptive numerical method for solution of the time-dependent Schrödinger equation is presented. By using an interpolating wavelet transform the number of used points can be reduced, constructing an adaptive sparse point representation. Two di erent discretisation approaches are studied in 1D ; one point-based and one based on equidistant blocks. Due to the nature of the wavelet transform a s...

We prove sharp Strichartz estimates for the semi-classical Schrödinger equation on a compact Riemannian manifold with smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrödinger equation outside a strictly convex obstacle, local existence for the H1-critical (quintic) Schrödinger equation and scattering for the sub-critical Schrödinger equation i...

In this paper, we study the scattering theory for the coupled KleinGordon-Schrödinger equation (the KGS equation) with the Yukawa type interaction, which is the certain quadratic interaction, in two space dimensions. It is well-known that for the two dimensional decoupled nonlinear Schrödinger and Klein-Gordon equations with the typical nonlinearity of the form |u|p−1u, there exist wave operato...

In this paper, the generalized sub-equation method is extended to investigate localized nonlinear waves of the one-dimensional nonlinear Schrödinger equation (NLSE) with potentials and nonlinearities depending on time and on spatial coordinates. With the help of symbolic computation, three families of analytical solutions of this NLS-type equation are presented. Based on these solutions, period...

We apply the proper orthogonal decomposition (POD) to the nonlinear Schrödinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic midpoint rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations...