Exponential Stability of a Schrödinger Equation Through Boundary Coupling a Wave Equation

2020 ◽  
Vol 65 (7) ◽  
pp. 3136-3142
Author(s):  
Jun-Min Wang ◽  
Fei Wang ◽  
Xiang-Dong Liu
Keyword(s):  
Wave Equation ◽  
Schrödinger Equation ◽  
Exponential Stability ◽  
Schrodinger Equation

Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves

2019 ◽  
Vol 33 (24) ◽  
pp. 1950285
Author(s):  
Saviour Worlanyo Akuamoah ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Wave Equation ◽  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Momentum Operator ◽  
Matter Wave ◽  
Relativistic Wave Equation ◽  
Relativistic Energy ◽  
Substitution Method ◽  
Energy And Momentum

In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.

scholarly journals Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold

2020 ◽  
Vol 10 (1) ◽  
pp. 569-583
Author(s):  
Fengyan Yang ◽  
Zhen-Hu Ning ◽  
Liangbiao Chen
Keyword(s):  
Riemannian Manifold ◽  
Schrödinger Equation ◽  
Exponential Stability ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Compact Riemannian Manifold ◽  
Nonlinear Schrodinger Equation ◽  
Point Of View ◽  
Physical Point ◽  
Nonlinear Schrödinger

Abstract In this paper, we consider the following nonlinear Schrödinger equation: $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g) is a smooth complete compact Riemannian manifold of dimension n(n = 2, 3) without boundary. For the damping terms −a(x)(1 − Δ)−1a(x)ut and $\begin{array}{} \displaystyle ia(x)(-{\it\Delta})^{\frac12}a(x)u, \end{array}$ the exponential stability results of system (0.1) have been proved by Dehman et al. (Math Z 254(4): 729-749, 2006), Laurent. (SIAM J. Math. Anal. 42(2): 785-832, 2010) and Cavalcanti et al. (Math Phys 69(4): 100, 2018). However, from the physical point of view, it would be more important to consider the stability of system (0.1) with the damping term ia(x)u, which is still an open problem. In this paper, we obtain the exponential stability of system (0.1) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.

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