Asymptotic behaviour of solutions of the difference Schrödinger equation

2011 ◽  
Vol 17 (11) ◽  
pp. 1555-1579 ◽  
Author(s):  
Vladimir Burd ◽  
Pavel Nesterov
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Schrodinger Equation ◽  
Asymptotic Behaviour Of Solutions ◽  
The Difference

scholarly journals Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

2009 ◽  
Vol 465 (2111) ◽  
pp. 3341-3360 ◽  
Author(s):  
Guillaume Michel Dujardin
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Boundary Value Problems ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

2012 ◽  
Vol 125 (1) ◽  
pp. 159-172 ◽  
Author(s):  
Vanilde Bisognin ◽  
Celene Buriol ◽  
Marcio V. Ferreira ◽  
Mauricio Sepúlveda ◽  
Octavio Vera
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Moving Boundaries ◽  
Nonlinear Schrödinger

scholarly journals COMPLEXIFIER VERSUS FACTORIZATION AND DEFORMATION METHODS FOR GENERATION OF COHERENT STATES OF A 1D NLHO I: MATHEMATICAL CONSTRUCTION

2013 ◽  
Vol 10 (10) ◽  
pp. 1350056 ◽  
Author(s):  
R. ROKNIZADEH ◽  
H. HEYDARI
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Jacobi Polynomials ◽  
One Dimensional ◽  
The Difference ◽  
Mathematical Construction

Three methods: complexifier, factorization and deformation, for construction of coherent states are presented for one-dimensional nonlinear harmonic oscillator (1D NLHO). Since by exploring the Jacobi polynomials [Formula: see text], bridging the difference between them is possible, we give here also the exact solution of Schrödinger equation of 1D NLHO in terms of Jacobi polynomials.

scholarly journals Determination of a Control Parameter for the Difference Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mesut Urun
Keyword(s):  
Schrödinger Equation ◽  
Difference Scheme ◽  
Schrodinger Equation ◽  
Control Parameter ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameterp,i(du(t)/dt)+Au(t)+iu(t)=f(t)+p,0<t<T,u(0)=φ,u(T)=ψ, in a Hilbert spaceHwith self-adjoint positive definite operatorAis constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.

scholarly journals About calculation of the ground-state energy of the helium atom

2021 ◽  
Vol 2067 (1) ◽  
pp. 012002
Author(s):  
E V Baklanov ◽  
P V Pokasov ◽  
A V Taichenachev
Keyword(s):  
Perturbation Theory ◽  
Numerical Calculation ◽  
Schrödinger Equation ◽  
Helium Atom ◽  
Ground State Energy ◽  
Ground State ◽  
Schrodinger Equation ◽  
State Energy ◽  
Calculation Results ◽  
The Difference

Abstract Two versions of the numerical calculation of the ground state energy of the helium atom are compared. First, the nonrelativistic Schrödinger equation with a fixed nucleus is solved, and then the perturbation theory is used. Another version solves this problem exactly. Comparison shows that the difference between the calculation results is 94 kHz.

On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation

1993 ◽  
Vol 123 (4) ◽  
pp. 763-782 ◽  
Author(s):  
Russell Johnson ◽  
Xingbin Pan
Keyword(s):  
Elliptic Equation ◽  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Radial Solutions ◽  
Nonlinear Schrödinger ◽  
Non Linear

SynopsisThis paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon potentials with large changes

2015 ◽  
Vol 93 (11) ◽  
pp. 1227-1234 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Kevin Malmgren
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Time Dependent ◽  
Step Potential ◽  
Localized State ◽  
Square Well ◽  
The Difference ◽  
Transmission And Reflection

We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

An initial boundary-value problem for the Schrödinger equation with long-range potential

1988 ◽  
Vol 417 (1853) ◽  
pp. 353-362 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Whittaker Functions ◽  
Wave Operators ◽  
Reference Problem ◽  
Initial Boundary Value

We consider the exterior Dirichlet problem in R 3 for the Schrödinger equation with a Coulomb potential. For such a Schrödinger equation defined in all of R 3 , the so-called reference problem, a fundamental solution is known in terms of Whittaker functions and the appropriate radiation conditions and the asymptotic behaviour of solutions can easily be obtained. Consequently, we can prove the limiting absorption principle, establish the existence of solutions to exterior boundary-value problems and show that the associated wave operators exist. This then enables questions concerning the asymptotic behaviour of time depen­dent solution of the exterior problem to be resolved in terms of the reference problem.

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