scholarly journals The exact WKB for cosmological particle production

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Seishi Enomoto ◽  
Tomohiro Matsuda
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Schrodinger Equation ◽  
Particle Production ◽  
Special Function ◽  
Scattering Problem ◽  
Turning Points ◽  
Bogoliubov Transformation ◽  
Zener Model ◽  
Stokes Phenomena

Abstract The Bogoliubov transformation in cosmological particle production can be explained by the Stokes phenomena of the corresponding ordinary differential equation. The calculation becomes very simple as far as the solution is described by a special function. Otherwise, the calculation requires more tactics, where the Exact WKB (EWKB) may be a powerful tool. Using the EWKB, we discuss cosmological particle production focusing on the effect of more general interaction and classical scattering. The classical scattering appears when the corresponding scattering problem of the Schrödinger equation develops classical turning points on the trajectory. The higher process of fermionic preheating is also discussed using the Landau-Zener model.

scholarly journals On Nonlinear Profile Decompositions and Scattering for an NLS–ODE Model

2018 ◽  
Vol 2020 (18) ◽  
pp. 5679-5722
Author(s):  
Scipio Cuccagna ◽  
Masaya Maeda
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hamiltonian System ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Golden Rule ◽  
Simplified Model ◽  
Radial Solutions ◽  
Fermi Golden Rule ◽  
Ode Model

Abstract In this paper, we consider a Hamiltonian system combining a nonlinear Schrödinger equation (NLS) and an ordinary differential equation. This system is a simplified model of the NLS around soliton solutions. Following Nakanishi [33], we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi’s framework and Fermi Golden Rule estimates on $L^4$ in time norms.

scholarly journals THE DIRICHLET PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION OF THE SECOND ORDER WITH THE OPERATOR OF DISTRIBUTED DIFFERENTIATION

2019 ◽  
pp. 48-57
Author(s):  
Б.И. Эфендиев
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dirichlet Problem ◽  
Special Function ◽  
Homogeneous Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Homogeneous Dirichlet Problem ◽  
Greens Function ◽  
The Green Function

В работе исследуется линейное обыкновенное дифференциальное уравнение второго порядка с оператором непрерывно распределенного дифференцирования, и для него изучается двухточечная краевая задача методом функции Грина. Вводится в рассмотрение специальная функция, в терминах которой строится функция Грина задачи Дирехле и доказываются основные свойства. Определены достаточные условия на ядро оператора непрерывно распределенного дифференцирования, гарантирующие выполнения условия разрешимости задачи Дирихле. В случае, когда однородная задача Дирихле для рассматриваемого однородного уравнения имеет нетривиальное решение получено неравенство типа Ляпунова для ядра оператора непрерывно распределенного дифференцирования. In this paper, we study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation, and for him we study the two-point boundary value problem by the Greens function method. A special function is introduced, in terms of which the Green function of the Direchle problem is constructed and the main properties are proved. Sufficient conditions on the kernel of the operator of continuously distributed differentiation are determined that guarantee the fulfillment of the solvability condition for the Dirichlet problem. In the case when the homogeneous Dirichlet problem for the homogeneous equation under consideration has a nontrivial solution, an analog of the Lyapunov inequality is obtained for the kernel of a continuously distributed ifferentiation operator.

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