scholarly journals Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 761
Author(s):  
Călin-Ioan Gheorghiu
Keyword(s):  
Schrödinger Equation ◽  
Small Parameter ◽  
Error Estimation ◽  
Schrodinger Equation ◽  
Error Control ◽  
High Order ◽  
Order Of Approximation ◽  
Schrödinger Equations ◽  
Spectral Collocation ◽  
Integration Interval

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

scholarly journals ASYMPTOTIC STABILITY OF NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

2005 ◽  
Vol 17 (10) ◽  
pp. 1143-1207 ◽  
Author(s):  
ZHOU GANG ◽  
I. M. SIGAL
Keyword(s):  
Asymptotic Stability ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Initial Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Generalized Nonlinear Schrödinger Equation

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

1999 ◽  
Vol 123 (1-3) ◽  
pp. 1-6 ◽  
Author(s):  
I.V. Puzynin ◽  
A.V. Selin ◽  
S.I. Vinitsky
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Order ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Order Accuracy

MORAWETZ AND INTERACTION MORAWETZ ESTIMATES FOR A QUASILINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 09 (04) ◽  
pp. 613-639 ◽  
Author(s):  
ALESSANDRO SELVITELLA ◽  
YUN WANG
Keyword(s):  
Schrödinger Equation ◽  
Plasma Physics ◽  
Schrodinger Equation ◽  
Energy Space ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation

We extend the classical Morawetz and interaction Morawetz machinery to a class of quasilinear Schrödinger equations coming from plasma physics. As an application of our main results we ensure the absence of pseudosolitons in the defocusing case. Our estimates are the first step to a scattering result in the energy space for this equation.

scholarly journals Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential

2006 ◽  
Vol 2006 ◽  
pp. 1-7
Author(s):  
Guanggan Chen ◽  
Jian Zhang ◽  
Yunyun Wei
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Variational Calculus ◽  
Orbital Stability ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Unbounded Potential

This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential iϕt=−Δϕ+V(x)ϕ−μ|ϕ|p−1ϕ−λ|ϕ|q−1ϕ, x∈ℝN, t≥0, where μ>0, λ>0, and 1<p<q<1+4/N. The potential V(x) is bounded from below and satisfies V(x)→∞ as |x|→∞. From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.

scholarly journals AN UNCERTAINTY PRINCIPLE FOR SOLUTIONS OF THE SCHRÖDINGER EQUATION ON -TYPE GROUPS

2020 ◽  
pp. 1-16
Author(s):  
AINGERU FERNÁNDEZ-BERTOLIN ◽  
PHILIPPE JAMING ◽  
SALVADOR PÉREZ-ESTEVA
Keyword(s):  
Schrödinger Equation ◽  
Uncertainty Principle ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Nilpotent Lie Groups ◽  
Schrödinger Equations ◽  
Uncertainty Principles ◽  
Uniqueness Of Solutions ◽  
Free Case ◽  
Formula Type

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$ -type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

A high order method for the solution of the single channel Schrödinger equation

1985 ◽  
Vol 54 (3) ◽  
pp. 733-740 ◽  
Author(s):  
Paul L. Devries
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Single Channel ◽  
High Order ◽  
Order Method ◽  
High Order Method
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