scholarly journals Non-relativistic and relativistic scattering by short-range potentials

2011 ◽  
Vol 369 (1939) ◽  
pp. 1228-1244 ◽  
Author(s):  
H. Arnbak ◽  
P. L. Christiansen ◽  
Yu. B. Gaididei
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Short Range ◽  
Perturbation Technique ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Relativistic Limit ◽  
Scattering Coefficients ◽  
Relativistic Scattering

Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the δ ′ potential in the Schrödinger equation and δ potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

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 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].

Solitary Waves for Quasilinear Schrödinger Equations Arising in Plasma Physics

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
João Marcos do Ó ◽  
Abbas Moameni
Keyword(s):  
Schrödinger Equation ◽  
Plasma Physics ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation

AbstractWe study the quasilinear Schrödinger equationizwhere W : ℝ

scholarly journals Schrödinger equations in noncylindrical domains: exact controllability

2006 ◽  
Vol 2006 ◽  
pp. 1-29
Author(s):  
G. O. Antunes ◽  
M. D. G. da Silva ◽  
R. F. Apolaya
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Schrodinger Equations ◽  
Lateral Boundary ◽  
Schrödinger Equations ◽  
Orthogonal Matrices ◽  
Noncylindrical Domain ◽  
Bounded Set

We consider an open bounded setΩ⊂ℝnand a family{K(t)}t≥0of orthogonal matrices ofℝn. SetΩt={x∈ℝn;x=K(t)y,for all y∈Ω}, whose boundary isΓt. We denote byQ^the noncylindrical domain given byQ^=∪0<t<T{Ωt×{t}}, with the regular lateral boundaryΣ^=∪0<t<T{Γt×{t}}. In this paper we investigate the boundary exact controllability for the linear Schrödinger equationu′−iΔu=finQ^(i2=−1),u=wonΣ^,u(x,0)=u0(x)inΩ0, wherewis the control.

A Sign-Changing Solution for an Asymptotically Linear Schrödinger Equation

2015 ◽  
Vol 58 (3) ◽  
pp. 697-716 ◽  
Author(s):  
Liliane A. Maia ◽  
Olimpio H. Miyagaki ◽  
Sergio H. M. Soares
Keyword(s):  
Variational Principle ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ekeland Variational Principle ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Asymptotically Linear ◽  
Deformation Lemma

AbstractThe aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

RELATED OPERATORS AND EXACT SOLUTIONS OF SCHRÖDINGER EQUATIONS

1998 ◽  
Vol 13 (28) ◽  
pp. 4913-4929 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
A. M. PERELOMOV ◽  
M. F. RAÑADA
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Meromorphic Function ◽  
Schrödinger Equation ◽  
Exact Solutions ◽  
Riccati Equation ◽  
Physical Interpretation ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Schrödinger Equations

We consider the Schrödinger equation just as a differential equation, disregarding the physical interpretation associated with solutions. By introducing the notion of A-related equations, A being a differential operator, we associate with it a Riccati equation and study the solutions when the potential is a meromorphic function.

scholarly journals Schrödinger Equations in Electromagnetic Fields: Symmetries and Noncommutative Integration

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1527
Author(s):  
Alexey Anatolievich Magazev ◽  
Maria Nikolaevna Boldyreva
Keyword(s):  
Schrödinger Equation ◽  
Electromagnetic Fields ◽  
Schrodinger Equation ◽  
Symmetry Algebra ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exact Integration ◽  
First Order ◽  
Symmetry Properties ◽  
Noncommutative Integration

We study symmetry properties and the possibility of exact integration of the time-independent Schrödinger equation in an external electromagnetic field. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra central extensions. Based on the well-known classification of the subalgebras of the algebra e(3), we classify all electromagnetic fields for which the corresponding time-independent Schrödinger equations admit first-order symmetry algebras. Moreover, we select the integrable cases, and for physically interesting electromagnetic fields, we reduced the original Schrödinger equation to an ordinary differential equation using the noncommutative integration method developed by Shapovalov and Shirokov.

Quasilinear Schrödinger Equations with Asymptotically Linear Nonlinearities

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Marcelo F. Furtado ◽  
Edcarlos D. Silva ◽  
Maxwell L. Silva
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonzero Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Asymptotically Linear ◽  
Positive Potential ◽  
Quasilinear Schrödinger Equation

AbstractWe deal with the existence of nonzero solution for the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere V is a positive potential and the nonlinearity g(x, s) behaves like K

scholarly journals Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation

2011 ◽  
Vol 369 (1944) ◽  
pp. 2155-2163 ◽  
Author(s):  
Denis Lapitski ◽  
Paul J. Dellar
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Lattice Boltzmann ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Relativistic Limit ◽  
Quantum Lattice ◽  
Lattice Boltzmann Scheme

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.

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