A Borel-Pompeiu Formula in ℂn and Its Application to Inverse Scattering Theory
01 Jan 2000-pp 117-134
TL;DR: In this article, a Borel-Pompeiu formula for functions in several complex variables using Clifford analysis was developed and used for a new inverse scattering transform in multidimensions.
Abstract: We develop a Borel-Pompeiu formula for functions in several complex variables using Clifford analysis. The obtained formula contains the BochnerMartinelli formula and additional information. The Borel-Pompeiu formula will be used for a new inverse scattering transform in multidimensions.
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TL;DR: In this article, a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics are presented, including a bibliography for the use of quaternions in physics.
Abstract: This is part one of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1 - Alphabetical bibliography, 2 - Analytical bibliography, 3 - Notations and terminology, and 4 - Formulas and identities. This quaternion bibliography will be further updated and corrected if necessary by the authors, who welcome any comment and reference that is not contained within the list.
43 citations
01 Jan 2010
TL;DR: The Dirac operator on the Euclidean space ℝ n, n ≥ 2, is a first-order differential operator with coefficients in the real Banach algebra associated with Ω n that has the defining property that it is a Dirac pair of differential operators, and every pair with the latter property is called a semi-Dirac pair as discussed by the authors.
Abstract: The Dirac operator on the Euclidean space ℝ n , n ≥ 2, is a first-order differential operator \( \mathfrak{D}_{euc,n} \) with coefficients in the real Clifford algebra \( \mathfrak{A}_{euc,n} \) associated with ℝ n that has the defining property \( \mathfrak{D}_{euc,n}^2 = - \Delta _{euc,n} \), where Δeuc,n stands for the standard Laplace operator on ℝ n . As generalizations of this class of operators, we investigate pairs (\( \mathfrak{D},\mathfrak{D}^\dag \)) of first-order homogeneous differential operators on ℝ n with coefficients in a real Banach algebra \( \mathfrak{A} \), such that \( \mathfrak{D}\mathfrak{D}^\dag = \mu _L \Delta _{euc,n} \) and \( \mathfrak{D}^\dag \mathfrak{D} = \mu _R \Delta _{euc,n} \), or \( \mathfrak{D}^\dag \mathfrak{D} + \mathfrak{D}^\dag \mathfrak{D} = \mu \Delta _{euc,n} \), where μL, μR, or μ are some elements of \( \mathfrak{A} \). Every pair (\( \mathfrak{D},\mathfrak{D}^\dag \)) that has the former property is called a Dirac pair of differential operators, and every pair (\( \mathfrak{D},\mathfrak{D}^\dag \)) with the latter property is called a semi-Dirac pair. Our goal is to prove that for any Dirac, or semi-Dirac pair, (\( \mathfrak{D},\mathfrak{D}^\dag \)), there are two interrelated Cauchy-Pompeiu type, and, respectively, two Bochner-Martinelli-Koppelman type integral representation formulas, one for \( \mathfrak{D} \) and another for \( \mathfrak{D}^\dag \). In addition, we show that the existence of such integral representation formulas characterizes the two classes of pairs of differential operators.
3 citations
TL;DR: In this article, a multidimensional ∂¯ method based on Clifford analysis is proposed to solve the inverse scattering problem for the n-dimensional time-dependent Schrodinger equations.
Abstract: The Schrodinger equation is one of the most important equations in mathematics, physics and also engineering. We outline some connections between transformations of non-linear equations, the Schrodinger equation and the inverse scattering transform. After some remarks on generalizations into higher dimensions we present a multidimensional ∂¯ method based on Clifford analysis. To explain the method we consider the formal solution of the inverse scattering problem for the n-dimensional time-dependent Schrodinger equations given by A.I. Nachman and M.J. Ablowitz. Copyright © 2002 John Wiley & Sons, Ltd.
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TL;DR: In this paper, the existence of admissible data for the inverse scattering problem in three spatial dimensions was investigated and the Borel-Pompeiu formula was applied for functions in ℂn.
Abstract: We apply a Borel-Pompeiu formula for functions in ℂn to investigate the existence of admissible data for the inverse scattering problem in three spatial dimensions.
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31 Jan 1992TL;DR: In this article, the authors bring together several aspects of soliton theory currently only available in research papers, including inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multidimensional space, and the ∂ method.
Abstract: Solitons have been of considerable interest to mathematicians since their discovery by Kruskal and Zabusky. This book brings together several aspects of soliton theory currently only available in research papers. Emphasis is given to the multi-dimensional problems arising and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the ∂ method. Thus, this book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.
4,198 citations
Book•
23 Jan 2012
TL;DR: In this paper, the boundary behavior of Cauchy integral functions is investigated and the results of the Schwarz Lemma are discussed, as well as the Zeros of Nevanlinna functions.
Abstract: Preliminaries.- The Automorphisms of B.- Integral Representations.- The Invariant Laplacian.- Boundary Behavior of Poisson Integrals.- Boundary Behavior of Cauchy Integrals.- Some Lp-Topics.- Consequences of the Schwarz Lemma.- Measures Related to the Ball Algebra.- Interpolation Sets for the Ball Algebra.- Boundary Behavior of H?-Functions.- Unitarily Invariant Function Spaces.- Moebius-Invariant Function Spaces.- Analytic Varieties.- Proper Holomorphic Maps.- The -Problem.- The Zeros of Nevanlinna Functions.- Tangential Cauchy-Riemann Operators.- Open Problems.
2,202 citations
Book•
[...]
01 Jan 1966
TL;DR: In this paper, the second edition of the Second Edition of Geometric Algebra, the authors present an approach to the interpretation of Clifford Algebra and the structure of Clifford algebras.
Abstract: Preface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure of Clifford Algebra.- 5.Reversion, Scalar Product.- 6.The Algebra of Space.- 7.The Algebra of Space-Time.- Part II:Electrodynamics.- 8.Maxwell's Equation.- 9.Stress-Energy Vectors.- 10.Invariants .- 11. Free Fields.- Part III:Dirac Fields.- 12.Spinors.- 13.Dirac's Equation.- 14.Conserved Currents.- 15.C, P, T.- Part IV:Lorentz Transformations.- 16.Reflections and Rotations.- 17.Coordinate Transformations.- 18.Timelike Rotations.- 19.Scalar Product.- Part V:Geometric Calculus.- 20.Differentiation.- 21.Coordinate Transformations.- 22.Integration.- 23.Global and Local Relativity.- 24.Gauge Transformation and Spinor Derivatives.- Conclusion.- Appendices.- A.Bases and Pseudoscalars.- B.Some Theorems.- C.Composition of Spacial Rotations.- D.Matrix Representation of the Pauli Algebra.
814 citations
Book•
01 Jan 1986
TL;DR: In this article, the origins of Geometric Algebra are discussed, including the development of geometric algebra and its application in physics, as well as its applications in geometry as physics.
Abstract: 1: Origins of Geometric Algebra.- 1-1. Geometry as Physics.- 1-2. Number and Magnitude.- 1-3. Directed Numbers.- 1-4. The Inner Product.- 1-5. The Outer Product.- 1-6. Synthesis and Simplification.- 1-7. Axioms for Geometric Algebra.- 2: Developments in Geometric Algebra.- 2-1. Basic Identities and Definitions.- 2-2. The Algebra of a Euclidean Plane.- 2-3. The Algebra of Euclidean 3-Space.- 2-4. Directions, Projections and Angles.- 2-5. The Exponential Function.- 2-6. Analytic Geometry.- 2-7. Functions of a Scalar Variable.- 2-8. Directional Derivatives and Line Integrals.- 3: Mechanics of a Single Particle.- 3-1. Newton's Program.- 3-2. Constant Force.- 3-3. Constant Force with Linear Drag.- 3-4. Constant Force with Quadratic Drag.- 3-5. Fluid Resistance.- 3-6. Constant Magnetic Field.- 3-7. Uniform Electric and Magnetic Fields.- 3-8. Linear Binding Force.- 3-9. Forced Oscillations.- 3-10. Conservative Forces and Constraints.- 4: Central Forces and Two-Particle Systems.- 4-1. Angular Momentum.- 4-2. Dynamics from Kinematics.- 4-3. The Kepler Problem.- 4-4. The Orbit in Time.- 4-5. Conservative Central Forces.- 4-6. Two-particle Systems.- 4-7. Elastic Collisions.- 4-8. Scattering Cross Sections.- 5: Operators and Transformations.- 5-1. Linear Operators and Matrices.- 5-2. Symmetric and Skewsymmetric Operators.- 5-3. The Arithmetic of Reflections and Rotations.- 5-4. Transformation Groups.- 5-5. Rigid Motions and Frames of Reference.- 5-6. Motion in Rotating Systems.- 6: Many-Particle Systems.- 6-1. General Properties of Many-Particle Systems.- 6-2. The Method of Lagrange.- 6-3. Coupled Oscillations and Waves.- 6-4. Theory of Small Oscillations.- 6-5. The Newtonian Many Body Problem.- 7: Rigid Body Mechanics.- 7-1. Rigid Body Modeling.- 7-2. Rigid Body Structure.- 7-3. The Symmetrical Top.- 7-4. Integrable Cases of Rotational Motion.- 7-5. Rolling Motion.- 7-6. Impulsive Motion.- 8: Celestial Mechanics.- 8-1. Gravitational Forces, Fields and Torques.- 8-2. Perturbations of Kepler Motion.- 8-3. Perturbations in the Solar System.- 8-4. Spinor Mechanics and Perturbation Theory.- 9: Foundations of Mechanics.- 9-1. Models and Theories.- 9-2. The Zeroth Law of Physics.- 9-3. Generic Laws and Principles of Particle Mechanics.- 9-4. Modeling Processes.- Appendixes.- A Spherical Trigonometry.- B Elliptic Functions.- C Units, Constants and Data.- Hints and Solutions for Selected Exercises.- References.
567 citations