This book introduces an interesting and alternative way to design absorbing boundary conditions (ABCs) for quantum wave equations, basically the nonlinear Schrödinger equation. The focus of this book is the application of the phase space filter approach to derive accurate radiation conditions for Schrödinger equations.
Researchers who are interested in partial differential equations and mathematical physics might find this book appealing.
Author(s): Avy Soffer, Chris Stucchio, Minh-Binh Tran
Series: SpringerBriefs on PDEs and Data Science
Publisher: Springer
Year: 2023
Language: English
Pages: 144
City: Singapore
Preface
Acknowledgment
Contents
1 Introduction
1.1 The Open Boundary Problem
1.1.1 Filtering Approach
1.1.2 Method Accuracy
1.1.3 The Advantages of the Method
1.1.4 The Disadvantages of the Method
1.1.5 Multiscale Extension
1.2 Comparison to Other Methods
1.2.1 Absorbing Boundary Conditions
Exact Dirichlet-Neumann Maps for the Schrödinger Equation
Paradifferential Strategy
1.2.2 Absorbing Potentials and Perfectly Matched Layers
Absorbing Potentials
Perfectly Matched Layers
2 Definitions, Notations and A Brief Introduction to Frames
2.1 Definitions and Notations
2.2 A Brief Introduction to Frames
2.2.1 Windowed Fourier Transform
2.2.2 Localization of Phase Space
2.2.3 Distinguished Sets of Framelets, Framelet Functionals
3 Windowed Fourier Transforms and Space Phase Numerics
3.1 Basic Definitions and Properties
3.2 Characterizing the Dual Window
3.3 Localization of Phases and Spaces
3.4 Phase Space Numerics
4 Description of Time Dependent Phase Space Filters
4.1 An Introductory Example
4.2 The TDPSF Algorithm
4.3 Implementation of the Algorithm
4.4 An Intuition of the Algorithm
4.5 Possible Enhancements
4.6 Slow Waves Multiscale Resolution
5 A More Practical Discussion: How to Choose the Parameters
6 The Behavior of Gaussian Framelets Under the Free Flow
6.1 Error Estimates
6.2 Location of Each Framelet
7 Assumptions and Accuracy Estimates
7.1 Assumptions
7.2 Discussions on the Assumptions
7.3 The Algorithm
7.4 Choices of the Parameters
7.5 Accuracy Estimates
7.5.1 Local (1 Step) Error
7.5.2 Global Error Estimates
7.6 Remarks
7.6.1 Almost Optimality of the Estimates
7.6.2 Difference with the Dirichlet-to-Neumann Approach
7.6.3 The Ping Pong Phenomenon
7.6.4 Bounds on kinf
8 Discussions on the Assumptions
8.1 Stationary Potentials
8.2 Assumption 1
8.3 Assumption 2
8.4 Assumption 3
8.5 Assumption 4
8.6 Assumption 5
8.7 Assumption 6
9 Proof of Theorem 7.5.6
9.1 Estimates of E(t), Q(t)
9.2 Estimates of E ( t )
9.3 Slow Waves
9.4 Estimates of Q ( t )
10 Proof of Theorems 7.5.4 and 7.5.5
10.1 Outgoing Waves
10.2 Residual Waves
11 Numerical Experiments
11.1 Case 1: T+R=E
11.2 Case 2: T+R ≠E
11.3 On Assumption 4
11.3.1 A Coincidence
11.3.2 Motivation in Constructing the TDPSF
References
