A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with c...

Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad Overview The goal of this research was to examine the quantum mechanical phenomenon of tunneling with respect to the Schrödinger Equation. A quantum mechanical wave is said to “tunnel” when it travels (propagates) through a classically forbidden reg...

For the one-dimensional Schrödinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some possible generalizations and extensively discusses the higher-dimensional case. H. Giacomini et A. Mou...

In the present paper, the authors consider the Schrödinger operator H with the Coulomb potential defined in R3m, where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain app...

As an important model in quantum semiconductor devices, the Schrödinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for high frequency waves outperform the geometrical op...

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...