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New method for computation of discrete spectrum of radical Schrödinger operator

Ivan Úlehla, Miloslav Havlíček (1980)

Aplikace matematiky

A new method for computation of eigenvalues of the radial Schrödinger operator - d 2 / d x 2 + v ( x ) , x ≥ 0 is presented. The potential v ( x ) is assumed to behave as x - 2 + ϵ if x → 0 + and as x - 2 - ϵ if x → + ∞ , ϵ ≥ 0 . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function z ( x , ℵ ) . It is shown that the eigenvalues are the discontinuity points of the function z ( ∞ , ℵ ) . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison...

New method to solve certain differential equations

Kazimierz Rajchel, Jerzy Szczęsny (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.

Nonlinear Dirac equations.

Ng, Wei Khim, Parwani, Rajesh R. (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Nonlinear Schrödinger equation on four-dimensional compact manifolds

Patrick Gérard, Vittoria Pierfelice (2010)

Bulletin de la Société Mathématique de France

We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy...

Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system

Piero D'Ancona, Damiano Foschi, Sigmund Selberg (2007)

Journal of the European Mathematical Society

We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in 1 + 3 dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this article, we provide a priorierror estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the...

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

Eric Cancès, Rachida Chakir, Yvon Maday (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

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