Related papers: Numerical Solution of the 1D Schr\"odinger Equatio…
Given two spherically symmetric and short range potentials $V_0$ and V_1 for which the radial Schrodinger equation can be solved explicitely at zero energy, we show how to construct a new potential $V$ for which the radial equation can…
The Schr\"{o}dinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schr\"{o}dinger equation into a second order differential equation by using an appropriate coordinate transformation. The…
A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy periodic with the lattice contains the first reciprocal lattice vector G1 =…
We consider the $1d$ cubic nonlinear Schr\"odinger equation with an external potential $V$ that is non-generic. Without making any parity assumption on the data, but assuming that the zero energy resonance of the associated Schr\"odinger…
We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schr\"odinger equations in one space dimension.
We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating…
We solve a radial Schr\"odinger equation for the case of a multichannel square well plus an exponential potential in one of the channels. The solution is obtained by summing exactly the infinite terms of the perturbative series for the…
In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q\inL_{1}[0,1] and q_{n}=0 for n=0,-1,-2,..., where q_{n} are the Fourier coefficients of q with respect to the system…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
We derive a representation formula for the Weyl solution to the Schr\"odinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with…
We present a program to simulate the dynamics of a wave packet interacting with a time-dependent potential. The time-dependent Schr\"odinger equation is solved on a one-, two-, or three-dimensional spatial grid using the split operator…
Recently, the Asymptotic Iteration Method (AIM) was used to calculate the energy spectrum for a short rang three parameter central potential which was introduced by H. Bahlouli and A. D. Alhaidari. The S-orbital wave solution of the…
Starting from the 3D Gross-Pitaevskii equation we revisit the dimensional reduction to an effective one-dimensional wave-equation that describes the longitudinal dynamics of a Bose condensate in an axially-symmetric external potential.…
We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states.…
An embedding method for solving the time-dependent Schr\"odinger equation is developed using the Dirac-Frenkel variational principle. Embedding allows the time-evolution of the wavefunction to be calculated explicitly in a limited region of…
Two rectangular models described by the one-dimensional Schroedinger equation with sharply localized potentials are suggested. The potentials have a multi-layer thin structure being composed from adjacent barriers and wells. Their peculiar…
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate…
The numerical solution of a linear Schr\"odinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and…
We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general…
In this paper, we apply the Schwarz Waveform Relaxation (SWR) method to the one dimensional Schr{\"o}dinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schr{\"o}dinger equation with time…