Solving Linear Partial Differential Equations: Spectra

Length: 408 pages
Edition: 1
Language: English
Publisher: World Scientific Publishing Company
Publication Date: 2020-07-27
ISBN-10: 9811216304
ISBN-13: 9789811216305
Sales Rank: #0 (See Top 100 Books)

Partial differential equations arise in many branches of science and vary in many ways. No one method can be used to solve all of them, and only a tiny percentage have been solved. In this book, we consider the general linear partial differential equation of arbitrary order m. Even this involves more methods that are known. We ask a simple question: when can an equation be solved and how many solutions does it have? We find that the answer is surprising even for equations with constant coefficients. We begin with these equations, first finding conditions which allow us to solve and obtain a finite number of solutions. We show how to obtain those solutions by analyzing the structure of the equation very carefully. A substantial part of the book is devoted to this. Then we tackle the more difficult problem of considering equations with variable coefficients. A large number of such equations are solved by comparing them to equations with constant coefficients. In numerous applications in the sciences, students and researchers are required to solve such equations in order to get the answers that they need. In many cases, the basic scientific theory requires that the resulting partial differential equation has a solution, and they are required to know how many solutions exist. This book deals with such situations.<\/p>

Contents
Preface
1 The Importance of the Spectrum
	1.1 Introduction
	1.2 The basic theorem
	1.3 The minimal operator
	1.4 The maximal extension
	1.5 Closed extensions
	1.6 The resolvent set via multipliers
	1.7 Perturbation by a potential
	1.8 The essential spectrum
	1.9 Relative compactness
	1.10 Variable coefficient operators
	1.11 Elliptic operators
	1.12 Verifying properties
2 Functional Analysis
	2.1 Banach and Hilbert spaces
	2.2 Linear operators
	2.3 Fredholm operators
	2.4 The spectrum
	2.5 Interpolation
	2.6 Intermediate extensions
	2.7 Self-adjoint operators
	2.8 Notes, remarks and references
3 Function Spaces
	3.1 Functions on En
	3.2 Fourier transforms
	3.3 Lp multipliers
	3.4 The spaces Hs,p
	3.5 Notes, remarks and references
4 Partial Differential Operators
	4.1 Constant coefficient operators
	4.2 Operators with variable coefficients
	4.3 Elliptic operators
	4.4 Notes, remarks and references
5 General Lp Theory
	5.1 The minimal operator
	5.2 The maximal extension
	5.3 The spectrum of the minimal operator
	5.4 The case p = 2
	5.5 Examples
	5.6 Perturbation by a potential
	5.7 Notes, remarks and references
6 Relative Compactness
	6.1 Orientation
	6.2 P0-boundedness
	6.3 P0-compactness
	6.4 Comparison of operators
	6.5 The operator qQ0
	6.6 The adjoint of P0 + q
	6.7 Smooth coefficients in L2
	6.8 The case p = 1
	6.9 The case 1 < p ≤ ∞
	6.10 Notes, remarks and references
7 Elliptic Operators
	7.1 An improvement
	7.2 A condition for boundedness
	7.3 Bessel potentials
	7.4 Some new spaces
	7.5 The Sobolev case
	7.6 Another estimate
	7.7 A general inequality
	7.8 Some comparisons
	7.9 Relative bounds
	7.10 Elliptic operators
	7.11 Notes, remarks and references
8 Operators Bounded from Below
	8.1 Introduction
	8.2 Regularly accretive extensions
	8.3 Invariance of the essential spectrum
	8.4 Perturbation by an operator
	8.5 An illustration
	8.6 Essential spectrum bounded from below
	8.7 Strongly elliptic operators
	8.8 A strengthened version for elliptic operators
	8.9 Perturbation by a potential. Elliptic case
	8.10 Perturbation by an operator. Elliptic case
	8.11 Notes, remarks and references
9 Self-Adjoint Extensions
	9.1 Existence
	9.2 Extensions with special properties
	9.3 Intervals containing the essential spectrum
	9.4 Essentially self-adjoint operators
	9.5 Finite negative spectrum
	9.6 Bounded operators
	9.7 Some related spaces
	9.8 Notes, remarks and references
10 Second Order Operators
	10.1 Introduction
	10.2 Essential self-adjointness
	10.3 Some observations
	10.4 Comparison of operators
	10.5 Estimating the essential spectrum
	10.6 The quadratic form J(')
	10.7 Adding of spectra
	10.8 Separation of coordinates
	10.9 Clusters
	10.10 Notes, remarks and references
11 Quantum Mechanics
	11.1 The Schrodinger operator for a particle
	11.2 Two particle systems
	11.3 The existence of bound states
	11.4 Systems of N particles
	11.5 The Zeeman effect
	11.6 Stability
	11.7 Lowest point of the spectrum
	11.8 Notes, remarks and references
Bibliography
Index