PERTURBATION

Asymptotic approximations and differential equations for advanced mathematical solutions and insights. Asymptotic Solution of Algebraic and Transcendental Equations

Introduction to Asymptotic Approximations

Matched Asymptotic Expansions

Multiple Scales

The WKB and Related Methods

The Method of Homogenization

Introduction to Bifurcation and Stability

Taylor Series

Solution and Properties of Transition Layer Equations

Asymptotic Approximations of Integrals

Second-Order Difference Equations

DelayEquations

Introduction to the Perturbation Theory of Hamiltonian Systems

Hamiltonian Equations

Introduction to the KAM Theory

Splitting of Asymptotic Manifolds

The Separatrix Map

Width of the Stochastic Layer

The Continuous Averaging Method

The Anti-Integrable Limit

Hill’s Formula

Appendix

Many-Body Perturbation Theory for Ab Initio Nuclear Structure

Basics

Single-Configurational Many-Body Perturbation Theory

Perturbatively-Enhanced No-Core Shell Model

Symmetry-Broken Many-Body Perturbation Theory

Appendices

Mathematical Foundations of Quantum Field Theory and Perturbative String Theory

Foundations for Quantum Field Theory

Quantization of Field Theories

Two-Dimensional Quantum Field Theories

Nonlinear Singular Perturbation Phenomena: Theory and Applications

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

QUADRATIC SINGULAR PERTURBATION PROBLEMS

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

SINGULARLY PERTURBED SYSTEMS

EXAMPLES AND APPLICATIONS

NON-PERTURBATIVE FIELD THEORY

From Two-Dimensional Conformal Field Theory to QCD in Four Dimensions

PART I NON-PERTURBATIVE METHODS IN TWO-DIMENSIONAL FIELD THEORY

PART II TWO-DIMENSIONAL NON-PERTURBATIVE GAUGE DYNAM

PART III FROM TWO TO FOUR DIMENSIONS

From massless free scalar field to conformal field theories

PART I NON-PERTURBATIVE METHODS IN TWO-DIMENSIONAL FIELD THEORY

Conformal field theory

Theories invariant under affine current algebras

Wess–Zumino–Witten model and coset models

Solitons and two-dimensional integrable models

Bosonization

PART II TWO-DIMENSIONAL NON-PERTURBATIVE GAUGE DYNAMICS

Gauge theories in two dimensions – basics

Bosonized gauge theories

The ’t Hooft solution of 2d QCD

Mesonic spectrum from current algebra

DLCQ and the spectra of QCD with fundamental and adjoint fermions

The baryonic spectrum of multiflavor QCD2 in the strong coupling limit

Confinement versus screening

QCD2 , coset models and BRST quantization

Generalized Yang–Mills theory on a Riemann surface

PART III FROM TWO TO FOUR DIMENSIONS

Conformal invariance in four-dimensional field theories and in QCD

Integrability in four-dimensional gauge dynamics

Large N methods in QCD4

From 2d bosonized baryons to 4d Skyrmions

From two-dimensional solitons to four-dimensional magnetic monopoles

Instantons of QCD

Summary, conclusions and outlook

NON PERTURBATIVE QUANTUM FIELD THEORY

Mathematical Aspects and Applications

INTRODUCTION

Phase Transitions and Continuous Symmetry Breaking

PHASE TRANSITIONS, GOLDSTONE BOSONS AND TOPOLOGICAL SUPERSELECTION RULES

FERROMAGNETIC MODELS IN CLASSICAL STATISTICAL MECHANICS AND RELATIVISTIC BOSE QUANTUM FIELD THEORY: MAIN RESULTS

INFRARED (GAUSSIAN) DOMINATION AND THERMODYNAMIC LIMIT

PHASE TRANSITIONS AND SPONTANEOUS SYMMETRY BREAKING FOR THE CLASSICAL N-VECTOR MODELS (MODELS 1-3,6)

THE (£•!') | - QUANTUM FIELD MODEL

PHASE TRANSITIONS IN TWO DIMENSIONAL QUANTUM FIELD MODELS

QUANTUM CRYSTALS AND MORE ABOUT THE PEIERLS ARGUMENT

THE PURE PHASES (HARMONIC FUNCTIONS) OF GENERALIZED PROCESSES

SPONTANEOUSLY BROKEN AND DYNAMICALLY ENHANCED GLOBAL AND LOCAL SYMMETRIES

QUANTUM THEORY OF NON-LINEAR INVARIANT WAVE (FIELD) EQUATIONS

Outline of a general theory of super selection sectors in r e l a t i v i s t i c quantum f i e ld theory

Bosonization, Topological Solitons and Fractional Charges in Two-Dimensional Quantum Field Theory

QUANTUM SKYRMIONS

Variational Problems on Vector Bundles

Gauge Theories, including (the Infrared Problem in) Quantum Electrodynamics

Magnetic Monopoles and Charged States in Four-Dimensional, Abelian Lattice Gauge Theories

QUANTIZED GAUGE FIELDS

Crossover Between Strong and Weak Coupling in Sl)(2) Pure Yang-Mills Theory

Quantum Field Theory of Anyons

ON THE TRIVIALITY OF k<p4 d THEORIES AND THE APPROACH TO THE CRITICAL POINT IN ^ 4 DIMENSIONS

REGGE CALCULUS AND DISCRETIZED GRAVITATIONAL FUNCTIONAL INTEGRALS

THE STATISTICAL MECHANICS OF SURFACES

STATISTICS OF FIELDS, THE YANG-BAXTER EQUATION, AND THE THEORY OF KNOTS AND LINKS

STATISTICS AND MONODROMY IN TWO- AND THREE-DIMENSIONAL QUANTUM-FIELD THEORY

STRUCTURE OF (UNITARY) RATIONAL CONFORMAL FIELD THEORY

New Developments in Quantum Field Theory

TWO-DIMENSIONAL CONFORMAL FIELD THEORY AND THREE-DIMENSIONAL TOPOLOGY

Quantum Statistics and Locality

UNIVERSALITY IN QUANTUM HALL SYSTEMS

Perturbation Methods

Introduction

Straightforward Expansions and Sources of No n u n i fo r m i ty

The Method of Strained Coordinates

The Methods of Matched and Composite Asymptotic Expansions

Variation of Parameters and Methods of Averaging

The Method of Multiple Scales

Asymptotic Solutions of Linear Equations

Introduction

Parameter Perturbations

Coordinate Perturbations

Order Symbols and Gauge Functions

Asymptotic Expansions and Sequences

Convergent versus Asymptotic Series

Nonuniform Expansions

Elementary Operations on Asymptotic Expansions

Exercises

Straightforward Expansions and Sources of No n u n i fo r m i ty

Infinite Domains

A Small Parameter Multiplying the Highest Derivative

Type Change of a Partial Differential Equation

The Presence of Singularities

The Role of Coordinate Systems

Exercises

The Method of Strained Coordinates

The Method of Strained Parameters

Lighthill’s Technique

Temple's Technique

Renormalization Technique

Limitations of the Method of Strained Coordinates

Exercises

Variation of Parameters and Methods of Averaging

The Method of Matched Asymptotic Expansions

The Method of Composite Expansions

Exercises

Variation of Parameters

The Method of Averaging

Struble’s Technique

The Krylov-Bogoliubov-Mitropolski Technique

The Method of Averaging by Using Canonical Variables

Von Zeipel’s Procedure

Averaging by Using the Lie Series and Transforms

Averaging by Using Lagrangians

Exercises

The Method of Multiple Scales

Description of the Method

Applications of the Derivative-Expansion Method

The Two-Variable Expansion Procedure

Generalized Method

Exercises

Asymptotic Solutions of Linear Equations

Second-Order Differential Equations

Systems of First-Order Ordinary Equations

Turning Point Problems

Wave Equations

Exercises

Perturbation Methods and Semilinear Elliptic Problems on Rn

Examples and Motivations

Pertubation in Critical Point Theory

Bifurcation from the Essential Spectrum

Elliptic Problems on Rn with Subcritical Growth

Elliptic Problems with Critical Exponent

The Yamabe Problem

Examples and Motivations

Elliptic equations on Rn

Bifurcation from the essential spectrum

Semiclassical standing waves of NLS

Pertubation in Critical Point Theory

A review on critical point theory

Critical points for a class of perturbed functionals, I

Critical points for a class of perturbed functionals, II

Amore general case

Bifurcation from the Essential Spectrum

A first bifurcation result

A second bifurcation result

A problemarising in nonlinear optics

Elliptic Problems on Rn with Subcritical Growth

The abstract setting

Study of the Ker[I 0 (zξ)]

A first existence result

Another existence result

Elliptic Problems with Critical Exponent

The unperturbed problem

On the Yamabe-like equation

Further existence results

The Yamabe Problem

Basic notions and facts

Some geometric preliminaries

First multiplicity results

Existence of infinitely-many solutions

Nonlinear Schr¨odinger Equations

Necessary conditions for existence of spikes

Spikes at non-degenerate critical points of V

The general case: Preliminaries

A modified abstract approach

Study of the reduced functional

Singularly Perturbed Neumann Problems

Preliminaries

Construction of approximate solutions

The abstract setting

Proof of Theorem 9.1

Concentration at Spheres for Radial Problems

Concentration at spheres for radial NLS

Proof of Theorem 10.1

Other results

Concentration at spheres for (Nε)

Perturbation Methods for Differential Equations

Asymptotic Series and Expansions

Regular Perturbation Methods

The Method of Strained CoordinateslParameters

Method of Averaging

The Method of Matched Asymptotic Expansions

Method of Multiple Scales

Applications to Plasma Physics

Miscellaneous Perturbation Methods

Exercises

Perturbation Methods in Applied Mathematics

Introduction

Limit Process Expansions Applied to Ordinary Differential Equations

Multiple-Variable Expansion Procedures

Applications to Partial Differential Equations

Examples from Fluid Mechanics

Perturbation Methods in Fluid Mechanics

The Nature of Perturbation Theory

Some Regular Perturbation Problems

The Techniques of Perturbation Theory

Some Singular Perturbation Problems in Airfoil Theory

The Method of Matched Asynptotic Expansions

The Method of Strained Coordinates

Viscous Flow at High Reynolds Number

Viscous Flow at Low Reynolds Number

Some Inviscid Singular Perturbation Problems

Other Aspects of Perturbation Theory

NOTES

Perturbation Methods, Bifurcation Theory and Computer Algebra

LINDSTEDT'S METHOD

CENTER MANIFOLDS

NORMAL FORMS

TWO VARIABLE EXPANSION METHOD

AVERAGING

LIE TRANSFORMS

LIAPUNOV-SCHMIDT REDUCTION

APPENDIX - INTRODUCfION TO MACSYMA

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations

Geometrical Preliminaries

Differential Calculus of Boundary Perturbations

Examples Using the Implicit Function Theorem

Bifurcation Problems

The Transversality Theorem

Generic Perturbation of the Boundary

Boundary Operators for Second-Order Elliptic Equations

The Method of Rapidly-Oscillating Solutions

Appendix 1 Eigenvalues of the Laplacian in the Presence of SymmetryAppendix 1 Eigenvalues of the Laplacian in the Presence of Symmetry

Appendix 2 On Micheletti’s Metric Space

Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

Introduction

Superstring Actions in AdS5 × S5 and AdS4 × CP3 Spaces

Geometric Properties of Semiclassically Quantized Strings

“Exact” Semiclassical Quantization of Folded Spinning Strings

Towards Precision Holography for Latitude Wilson Loops

Light-Like Cusp Anomaly and the Interpolating Function in ABJM

AdS5 × S5 Superstring on the Lattice

Appendix A - Jacobi Elliptic Functions

Appendix B Methods for Functional Determinants in One Dimension

Appendix C Exact Spectrum for a Class of Fourth-Order Differential Operators

Appendix D Conventions forWorldsheet Geometry

Appendix E Details on the Null Cusp Fluctuation Lagrangian in AdS4 × CP3

Appendix F Simulating Strings on the Lattice

Perturbative Quantum Chromodynamics

Factorization of Hard Processes in QCD

Exclusive Processes in Quantum Chromodynamics

Coherence and Physics of QCD Jets

Pomeron in Quantum and Physics of QCD Jets

Infrared Singularities and Coherent States in Gauge Theories

Sudakov From Factors

Perturbative Quantum Electrodynamics and Axiomatic Field Theory

Part I What Is Quantum Electrodynamics?

Part II Perturbation Theory

Part III Particles and Their Reactions

What Is Quantum Electrodynamics?

Relativistic Covariance

Electrodynamics as a Classical Field Theory

The Basic Principles of Relativistic Quantum Field Theory

Free Fields

An Outline of Interacting QED

The Electric Charges

Perturbation Theory

The Program of Perturbation Theory

Unrenormalized Solution

Renormalization and the UV Problem

The IR Problem for Wightman Functions

Physical States

Particles and Their Reactions

The Standard View

Particle Probes

Interacting Particles

Reactions

Cross Sections

Singular Perturbation Methods for Ordinary Differential Equations

Examples Illustrating Regular and Singular Perturbation Concepts

Singularly Perturbed Initial Value Problems

Singularly Perturbed Boundary Value Problems

Appendix: The Historical Development of Singular Perturbations

Examples Illustrating Regular and Singular Perturbation Concepts

The Harmonic Oscillator: Low Frequency Situation

Introductory Definitions and Remarks

A Simple First-Order Linear Initial Value Problem

Some Second-Order Two-Point Problems

Regular Perturbation Theory for Initial Value Problems

Singularly Perturbed Initial Value Problems

A Nonlinear Problem from Enzyme Kinetics

The Solution of Linear Systems Using Transformation Methods

Inner and Outer Solutions of Model Problems

The Nonlinear Vector Problem (Tikhonov-Levinson Theory)

An Outline of a Proof of Asymptotic Correctness

Numerical Methods for Stiff Equations

Relaxation Oscillations

A Combustion Model

Linear and Nonlinear Examples of Conditionally Stable Systems

Singular Problems

Singularly Perturbed Boundary Value Problems

Second Order Linear Equations (without Turning Points)

Linear Scalar Equations of Higher Order

First-Order Linear Systems

An Application in Control Theory

Some Linear Turning Point Problems

Quasilinear Second-Order Problems

Existence, Uniqueness, and Numerical Computation of Solutions

Quasilinear Vector Problems

An Example with an Angular Solution

Nonlinear Systems

A Nonlinear Control Problem

Semiconductor Modeling

Shocks and Transition Layers

A Geometric Analysis for Some Autonomous Equations

Semilinear Problems

Singular Perturbation Methods in Control Analysis and Design

TIME-SCALE MODELING

LINEAR TIME-INVARIANT SYSTEMS

LINEAR FEEDBACK CONTROL

STOCHASTIC LINEAR FILTERING AND CONTROL

LINEAR TIME-VARYING SYSTEMS

OPTIMAL CONTROL

NONLINEAR SYSTEMS

TIME-SCALE MODELING

Introduction

The Standard Singular Perturbation Model

Time-Scale Properties of the Standard Model

Slow and Fast Manifolds

Construction of Approximate Models

From Nonstandard to Standard Forms

Case Studies in Scaling

Exercises

LINEAR TIME-INVARIANT SYSTEMS

Introduction

The Block-Triangular Forms

EIGENVALUE PROPERTIES

The Block-Diagonal Form: Eigenspace Properties

Validation of Approximate Models

Controllability and Observability

Frequency-Domain Models

Exercises

LINEAR FEEDBACK CONTROL

Introduction

Composite State-Feedback Control

Eigenvalue Assignment

Near-Optimal Regulators

A Corrected Linear-Quadratic Design

High-Gain Feedback

Robust Output-Feedback Design

Exercises

STOCHASTIC LINEAR FILTERING AND CONTROL

Introduction

Slow-Fast Decomposition in the Presence of White-Noise Inputs

The Steady-State Kalman-Bucy Filter

The Steady-State LOG Controller

An Aircraft Autopilot Case Study

Corrected LOG Design and the Choice of the Decoupling Transformation

Scaled White-Noise Inputs

Exercises

LINEAR TIME-VARYING SYSTEMS

Introduction

Slowly Varying Systems

Decoupling Transformation

Uniform Asymptotic Stability

Stability of a Linear Adaptive System

OPTIMAL CONTROL

Introduction

Boundary Layers in Optimal Control

The Reduced Problem

Near-Optimal Linear Control

Nonlinear and Constrained Control

Cheap Control and Singular Arcs

Exercises

Adiabatic Perturbation Theory in Quantum Dynamics

Introduction

First order adiabatic theory

Applications and extensions

Space-adiabatic perturbation theory

Quantum dynamics in periodic media

Adiabatic decoupling without spectral gap

Pseudodifferential operators

Operator-valued Weyl calculus for τ-equivariant symbols

Related approaches

List of symbols

Introduction

The time-adiabatic theorem of quantum mechanics

Space-adiabatic decoupling: examples from physics .

Outline of contents and some left out topics.

Molecular dynamics

The Dirac equation with slowly varying potentials

First order adiabatic theory

The classical time-adiabatic result

Perturbations of fibered Hamiltonians

Time-dependent Born-Oppenheimer theory: Part I .

Constrained quantum motion

Time-dependent Born-Oppenheimer theory: Part I

A global result

Local results and effective dynamics

The semiclassical limit: first remarks

Born-Oppenheimer approximation in a magnetic field and Berry’s connection

Constrained quantum motion

The classical problem

A quantum mechanical result

Comparison

Semiclassical limit for effective Hamiltonians

Space-adiabatic perturbation theory

Almost invariant subspaces

Mapping to the reference space

Effective dynamics

Semiclassical limit for effective Hamiltonian

Semiclassical analysis for matrix-valued symbols .

Geometrical interpretation: the generalized Berry connection

Semiclassical observables and an Egorov theorem

Applications and extensions

The Dirac equation with slowly varying potentials

Time-dependent Born-Oppenheimer theory: Part II .

The time-adiabatic theorem revisited

How good is the adiabatic approximation?

Semiclassical observables and an Egorov theorem

Back-reaction of spin onto the translational motion

The B.-O. approximation near a conical eigenvalue crossing

Decoupling electrons and positrons

Semiclassical limit for electrons: the T-BMT equation

Back-reaction of spin onto the translational motion

Quantum dynamics in periodic media

The periodic Hamiltonian

Adiabatic perturbation theory for Bloch bands

The almost invariant subspace

The intertwining unitaries

The effective Hamiltonian

Semiclassical dynamics for Bloch electrons

Adiabatic decoupling without spectral gap

Time-adiabatic theory without gap condition

Space-adiabatic theory without gap condition

Effective N-body dynamics in the massless Nelson model

Formulation of the problem

Mathematical results

Weyl quantization and symbol classes

Composition of symbols: the Weyl-Moyal product

Pseudodifferential operators

Advanced Mathematical Methods for Scientists and Engineers I

Asymptotic Methods and Perturbation Theory

Ordinary Differential Equations

Difference Equations

Approximate Solution of Difference Equations

Asymptotic Expansion of Integrals

PART I FUNDAMENTALS

Asymptotic Expansion of Integrals

PART II LOCAL ANALYSIS

Approximate Solution of Linear Differential Equations

Approximate Solution of Nonlinear Differential Equations

PART III PERTURBATION METHODS

Perturbation Series

Summation of Series

PART IV GLOBAL ANALYSIS

Boundary Layer Theory

WKB Theory

Multiple-Scale Analysis

Appendix-Useful Formulas

Ordinary Differential Equations

Ordinary Differential Equations

Initial-Value and Boundary-Value Problems

Theory of Homogeneous Linear Equations

Solutions of Homogeneous Linear Equations

Inhomogeneous Linear Equations

First-Order Nonlinear Differential Equations

Higher-Order Nonlinear Differential Equations

Eigenvalue Problems

Differential Equations in the Complex Plane

Problems for Chapter 1

Difference Equations

The Calculus of Differences

Elementary Difference Equations

Homogeneous Linear Difference Equations

Inhomogeneous Linear Difference Equations

Nonlinear Difference Equations

Problems for Chapter 2

Approximate Solution of Linear Differential Equations

Classification of Singular Points of Homogeneous Linear Equations

Local Behavior Near Ordinary Points of Homogeneous Linear Equations

Local Series Expansions About Regular Singular Points of Homogeneous Linear Equations

Local Behavior at Irregular Singular Points of Homogeneous Linear Equations

Irregular Singular Point at Infinity

Local Analysis of Inhomogeneous Linear Equations

Asymptotic Relations

Asymptotic Series

Problems for Chapter 3

Approximate Solution of Nonlinear Differential Equations

Spontaneous Singularities

Approximate Solutions of First-Order Nonlinear Differential Equations

Approximate Solutions to Higher-Order Nonlinear Differential Equations

Nonlinear Autonomous Systems

Higher-Order Nonlinear Autonomous Systems

Problems for Chapter 4

Approximate Solution of Difference Equations

Spontaneous Singularities

Ordinary and Regular Singular Points of Linear Difference Equations

Local Behavior Near an Irregular Singular Point at Infinity: Determination of Controlling Factors

Asymptotic Behavior of n! as n →∞ : The Stirling Series

Local Behavior Near an Irregular Singular Point at Infinity: Full Asymptotic Series

Local Behavior of Nonlinear Difference Equations

Problems for Chapter 5

Approximate Expansion of Integrals

Introduction

Elementary Examples

Integration by Parts

Laplace's Method and Watson's Lemma

Method of Stationary Phase

Method of Steepest Descents

Asymptotic Evaluation of Sums

Problems for Chapter 6

Perturbation Series

Perturbation Theory

Regular and Singular Perturbation Theory

Perturbation Methods for Linear Eigenvalue Problems

Asymptotic Matching

Mathematical Structure of Perturbative Eigenvalue Problems

Problems for Chapter 7

Summation Series

Improvement of Convergence

Summation of Divergent Series

Pade Summation

Continued Fractions and Pade Approximants

Convergence of Pade Approximants

Pade Sequences for StieItjes Functions

Problems for Chapter 8

Boundary Layer Theory

Introduction to Boundary-Layer Theory

Mathematical Structure of Boundary Layers

Distinguished Limits and Boundary Layers of Thickness # e

Miscellaneous Examples of Linear Boundary-Layer Problems

Internal Boundary Layers

Nonlinear Boundary-Layer Problems

Problems for Chapter 9

WKB Theory

The Exponential Approximation for Dissipative and Dispersive Phenomena

Conditions for Validity of the WKB Approximation

Patched Asvmptotic Approximations: WKB Solution of Inhomogeneous Linear Equations

Matched Asymptotic Approximations: Solution of the One-Turning-Point Problem

Two-Turning-Point Problems: Eigenvalue Condition

Tunneling

Brief Discussion of Higher-Order WKB Approximations

Problems for Chapter 10

Multiple-Scale Analysis

Resonance and Secular Behavior

Multiple-Scale Analysis

Examples of Multiple-Scale Analysis

The Mathieu Equation and Stability

Problems for Chapter 11

Appendix-Useful Formulas

AIRY FUNCTIONS

MODIFIED BESSEL FUNCTIONS

BESSEL FUNCTIONS

PARABOLIC CYLINDER FUNCTIONS

GAMMA AND DIGAMMA (PSI) FUNCTIONS

EXPONENTIAL INTEGRALS

AIRY FUNCTIONS

Differential equation:

Taylor series:

Functional relations:

Relation to Bessel functions:

Asymptotic expansions:

Integral representations:

MODIFIED BESSEL FUNCTIONS

Differential equation:

Frobenius series:

Functional relations:

Asymptotic expansions:

Integral representations:

Difference equation:

Generating function

BESSEL FUNCTIONS

Differential equation:

Frobenius series:

Functional relations:

Other differential equations:

Asymptotic expansions:

Integral representations:

Difference equation:

Generating function

PARABOLIC CYLINDER FUNCTIONS

Differential equation:

Taylor series:

Functional relations:

Asymptotic expansions:

Integral representations:

Difference equation:

Relation to Hermite polynomials

GAMMA AND DIGAMMA (PSI) FUNCTIONS

Integral representations:

Difference equation:

Special values:

Stirling's asymptotic formula:

Other formulas:

Psi function:

Taylor series:

Asymptotic expansions:

EXPONENTIAL INTEGRALS

Integral representations:

Series expansion

Recurrence relations:

Asymptotic expansions:

Algebraic Methods in Nonlinear Perturbation Theory***

Matrix Perturbation Theory

Systems of Ordinary Differential Equations with a Small Parameter

Examples

Reconstruction

Equations in Partial Derivatives

Matrix Perturbation Theory

Perturbation Theory for a Linear Operator

Diagonal Leading Operator

Nilpotent Leading Operator. The Reconstruction Problem

Main Formulas

General Case. The Normal Form of the Matrix of the Operator M

Systems of Ordinary Differential Equations with a Small Parameter

Passage to the Linear Problem. Change of Variables Operator

General Formulation of the Perturbation Theory Problem

Canonical Form of First Order Operator Xo

An Algebraic Formulation of the Perturbation Theory Problem

The Normal Form of an Operator with Respect to a Nilpotent Xo. The Reconstruction Problem

A Connection with N. N. Bogolyubov's Ideas

The Motion Near the Stationary Manifold

Hamiltonian Systems

Examples

The Pendulum of Variable Length

A Second Order Linear Equation

P. L. Kapitsa's Problem: A Pendulum Suspended from an Oscillating Point

Van der Pol Oscillator with Small Damping

Duffing Oscillator

Drift of a Charged Particle in an Electromagnetic Field . . . .

Nonlinear System: Example of an Extension of an Operator

Nonlinear Oscillator with Small Mass and Damping

Reconstruction

Introduction

New Leading Operators in the First Type Problems

The Second Type Problems. "Algebraic" Method of Reconstruction

"Thajectory" Method of Reconstruction

Matching

Example: Illustration for 4.5

Example: Appearance of a New Singularity

Example: Passing Through a Resonance

Example: WKB-Type Problem

Example: Lighthill's Problem

Example: Singularity of Coefficients of an Operator

Example: A Second Order Linear Equation

Example: Van der Pol Oscillator (Relaxation Oscillations)

Equations in Partial Derivatives

Functional Derivatives

Equations with Partial Derivatives Whose Principal Part Is an Ordinary Differential Equation ...

Partial Derivatives. On Whitham Method

Geometric Optics and the Maslov Method

Problem (Whitham)

Problem. Diffraction of Short Waves on a Circle (Semishade)

One-Dimensional Shock Wave

Example: Solving the Perturbed Two-Body Initial Value Prob-

lemfor Hamilton’s Principal Function: Applications to Boundary Value Problems

Introduction

Hamiltonian Dynamics

Perturbation Theory for Hamilton’s Principal Function

Perturbation Theory for the Two-Point Boundary Value Prob- lem

Perturbation Theory for the Initial Value Problem

Perturbed Rotating Two-Body Problem

Example: Solving the Perturbed Two-Body Two-Point Bound- ary Value Problem

Example: Solving the Perturbed Two-Body Initial Value Prob- lem

Introduction and Motivation

Analytic Perturbation Theory and Its Applications

Finite Dimensional Perturbations

Applications to Optimization and Markov Processes

Infinite Dimensional Perturbations

Introduction and Motivation

Background

Raison d’Être and Exclusions

Organization of the Material

Possible Courses with Prerequisites

Inversion of Analytically Perturbed Matrices

Introduction and Preliminaries

Inversion of Analytically Perturbed Matrices: Algebraic Approach

Problems

Perturbation of Null Spaces, Eigenvectors, and Generalized Inverses

Introduction

Perturbation of Null Spaces and the Eigenvalue Problem

Perturbation of Generalized Inverses: Complex Analytic Approach

Problems

Polynomial Perturbation of Algebraic Nonlinear Systems

Introduction

Preliminaries on Gröbner Bases and Buchberger’s Algorithm∗

Reduction of the System of Perturbed Polynomials

Classification of ExpansionTypes

Irreducible Factorization of Bivariate Polynomials

Computing Series Coefficients for Regularly Perturbed Polynomials

Newton Polygon Method for Singularly Perturbed Polynomials

An Example of Application to Optimization

Problems

Applications to Optimization

Introduction and Motivation

Asymptotic Simplex Method

Asymptotic Gradient Projection Methods

Asymptotic Analysis for General Nonlinear Programming: Complex Analytic Perspective

Problems

Applications to Markov Chains

Introduction, Motivation and Preliminaries

Asymptotic Analysis of Deviation, Fundamental, and Mean Passage Time Matrices

Google PageRank as a Perturbed Markov Chain

Problems

Applications to Markov Decision Processes

Markov Decision Processes: Concepts and Introduction

Problems

Nearly Completely Decomposable Markov Decision Processes

Parametric Analysis of Markov Decision Processes

Perturbed Markov Chains and the Hamiltonian Cycle Problem

Analytic Perturbation of Linear Operators

Introduction

Preliminaries from Finite Dimensional Theory

KeyExamples

Motivating Applications

Review of Banach and Hilbert Spaces

Inversion of Linearly Perturbed Operators on Hilbert Spaces

Inversion of Linearly Perturbed Operators on Banach Spaces

Polynomial andAnalyticPerturbations

Problems

Background on Hilbert Spaces and Fourier Analysis

The Hilbert Space L2([−π,π])

The Fourier Series Representation on ([−π,π]) .

Fourier Series Representation on L2([−π,π])

The Space l2

The Hilbert Space H1 0 ([−π,π])

Fourier Series in H1 0 ([−π,π]) .

The Complex Hilbert Space L2([−π,π])

Fourier Series in the Complex Space L2([−π,π])

The Hilbert Space L2()

The Fourier Integral Representation on 0(R)

The Fourier Integral Representation on L2(R)

The Hilbert Space H1 0 (R)

Fourier Integrals in H1 0 (R)

The Complex Hilbert Space L2(R)

Fourier Integrals in the Complex Space L2(R)

Asymptotic Analysis and Perturbation Theory

Introduction to Asymptotics

Asymptotics of Integrals

Speeding Up Convergence

Differential Equations

Asymptotic Series Solutions for Dierential Equations

Difference Equations

Perturbation Theory

WKBJ Theory

Multiple-Scale Analysis

Appendix{Guide to the Special Functions

Introduction to Asymptotics

Basic Definitions

Limits via Asymptotics

Asymptotics of Integrals

Inverse Functions

Dominant Balance

Asymptotics of Integrals

Integrating Taylor Series

Repeated Integration by Parts

Laplace's Method

Review of Complex Numbers

Method of Stationary Phase

Method of Steepest Descents

Speeding Up Convergence

Shanks Transformation

Richardson Extrapolation

Differential Equations

Classication of Differential Equations

First Order Equations

Taylor Series Solutions

Frobenius Method

Asymptotic Series Solutions for Differential Equations

Behavior for Irregular Singular Points

Full Asymptotic Expansion

Local Analysis of Inhomogeneous Equations

Local Analysis for Non-linear Equations

Difference Equations

Classification of Difference Equations

First Order Linear Equations

Analysis of Linear Difference Equations

The Euler-Maclaurin Formula

Taylor-like and Frobenius-like Series Expansions

Perturbation Theory

Introduction to Perturbation Theory

Regular Perturbation for Differential Equations

Singular Perturbation for Differential Equations

Asymptotic Matching

WKBJ Theory

The Exponential Approximation

Region of Validity

Turning Points

Multiple-Scale Analysis

Strained Coordinates Method (Poincare-Lindstedt)

The Multiple-Scale Procedure

Two-Variable Expansion Method

Appendix{Guide to the Special Functions

Asymptotic Perturbation Theory of Waves

Perturbed Oscillations and Waves: Introductory Examples

Perturbation Method for Quasi-Harmonic Waves

Perturbation Method for Non-Sinusoidal Waves

Nonlinear Waves of Modulation

Perturbation Methods for Solitary Waves and Fronts

Perturbed Solitons

Interaction and Ensembles of Solitons and Kinks

Dissipative and Active Systems. Autowaves

Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains

Boundary Value Problems for the Laplace Operator in Domains Perturbed Near Isolated Singularities

Dirichlet and Neumann Problems for the Laplace Operator in Domains with Corners and Cone Vertices

Dirichlet and Neumann Problems in Domains with Singularly Perturbed Boundaries

General Elliptic Boundary Value Problems in Domains Perturbed Near Isolated Singularities of the Boundary

Elliptic Boundary Value Problems in Domains with Smooth Boundaries, in a Cylinder, and in Domains with Cone Vertices

Asymptotics of Solutions to General Elliptic Boundary Value Problems in Domains Perturbed Near Cone Vertices

Variants and Corollaries of the Asymptotic Theory

Asymptotic Behaviour of Functionals on Solutions of Boundary Value Problems in Domains Perturbed Near Isolated Boundary Singularities

Asymptotic Behaviour of Energy Integrals for Small Perturbations of the Boundary Near Corners and Isolated Points

Asymptotic Behaviour of Energy Integrals for Particular Problems of Mathematical Physics

Asymptotic Behaviour of Eigenvalues of Boundary Value Problems in Domains with Small Holes

Asymptotic Expansions of Eigenvalues of Classic Boundary Value Problems

Homogeneous Solutions of Boundary Value Problems in the Exterior of a Thin Cone

Boundary Value Problems in Domains Perturbed Near Multidimensional Singularities of the Boundary

Boundary Value Problems in Domains with Edges on the Boundary

Asymptotics of Solutions to Classical Boundary Value Problems in a Domain with Thin Cavities

Asymptotics of Solutions to the Dirichlet Problem for High Order Equations in a Domain with a Thin Tube Excluded

Behaviour of Solutions of Boundary Value Problems in Thin Domains

The Dirichlet Problem in Domains with Thin Ligaments

Boundary Value Problems of Mathematical Physics in Thin Domains

General Elliptic Problems in Thin Domains

Elliptic Boundary Value Problems with Oscillating Coefficients or Boundary of Domain

Elliptic Boundary Value Problems with Rapidly Oscillating Coefficients

Paradoxes of Limit Passage in Solutions of Boundary Value Problems When Smooth Domains Are Approximated by Polygons

Homogenization of a Differential Operator on a Fine Periodic Net of Curves

Homogenization of Equations on a Fine Periodic Grid

List of Symbols

Basic Symbols

Symbols for function spaces and related concepts

Symbols for functions, distributions and related concepts

Other symbols

Canonical Perturbation Theories Degenerate Systems and Resonance

The Hamilton–Jacobi Theory

Angle–Action Variables. Separable Systems

Classical Perturbation Theories

Resonance

Lie Mappings

Lie Series Perturbation Theory

Non-Singular Canonical Variables

Lie Series Theory for Resonant Systems

Single Resonance near a Singularity

Nonlinear Oscillators

Bohlin Theory

The Simple Pendulum

Andoyer Hamiltonian with k = 1

Andoyer Hamiltonian with k ≥ 2

Canonical Pertubation Equations

Hamilton’s Principle

Canonical Transformations

Lagrange Brackets

Poisson Brackets

The Extended Phase Space

Gyroscopic Systems

The Partial Differential Equation of Hamilton and Jacobi

One-Dimensional Motion with a Generic Potential

Involution. Mayer’s Lemma. Liouville’s Theorem

Periodic Motions

Direct Construction of Angle–Action Variables

Actions in Multiperiodic Systems. Einstein’s Theory

Separable Multiperiodic Systems

Simple Separable Systems

The Separable Cases of Liouville and St¨ackel

Kepler Motion

Degeneracy

Angle–Action Variables of a Quadratic Hamiltonian

The Problem of Delaunay

The Poincar´e Theory

Averaging Rule

Degenerate Systems. The von Zeipel–Brouwer Theory

Small Divisors and Resonance

An Example – Part I

Linear Secular Theory

An Example – Part II

Iterative Use of von Zeipel–Brouwer Operations

Divergence of the Series. Poincar´e’s Theorem

Kolmogorov’s Theorem

Inversion of a Jacobian Transformation

Lindstedt’s Direct Calculation of the Series

The Method of Delaunay’s Lunar Theory

Introduction of the Square Root of the Small Parameter

Delaunay Theory According to Poincare´

Garfinkel’s Ideal Resonance Problem

Angle–Action Variables of the Ideal Resonance Problem

Morbidelli’s Successive Elimination of Harmonics

Lie Transformations

Lie Derivatives

Lie Series

Inversion of a Lie Mapping

Lie Series Expansions

Introduction

Lie Series Theory with Angle–Action Variables

Comparison to Poincar´e Theory. Example I

Comparison to Poincar´e Theory. Example II

Hori’s General Theory. Hori Kernel and Averaging

Topology and Small Divisors

Hori’s Formal First Integral

“Average” Hamiltonians

Singularities of the Actions

Poincar´e Non-Singular Variables

The d’Alembert Property

Regular Integrable Hamiltonians

Lie Series Expansions About the Origin

Lie Series Perturbation Theory in Non-Singular Variables

The Non-Resonance Condition

Bohlin’s Problem (The Single-Resonance Problem)

Outline of the Solution

Functions Expansions

Perturbation Equations

Averaging

An Example

Example with a Separated Hori Kernel

One Degree of Freedom

Resonances Near the Origin: Real and Virtual

One Degree of Freedom

Many Degrees of Freedom. One Single Resonance

Sessin Transformation and Integral

Quasiharmonic Hamiltonian Systems

Formal Solutions. General Case

Exact Commensurability of Frequencies (Resonance)

Birkhoff Normalization

The Restricted Three-Body Problem

The H´enon–Heiles Hamiltonian

Systems with Multiple Commensurabilities

Parametrically Excited Systems

Bohlin’s Resonance Problem

Bohlin’s Perturbation Equations

Poincar´e Singularity

An Extension of Delaunay Theory

Equations of Motion

Angle–Action Variables of the Pendulum

Small Oscillations of the Pendulum

Direct Construction of Angle–Action Variables

The Neighborhood of the Pendulum Separatrix

The Separatrix or Whisker Map

The Standard Map

Andoyer Hamiltonians

Centers and Saddle Points

Morphogenesis

Width of the Libration Zone

Small-Amplitude Librations

THE THEORY OF SINGULAR PERTURBATIONS

General Introduction

Asymptotic Expansions

Regular Perturbations .

The Method o f the Strained Coordinate

The Method o f Averaging

The Method of Multiple Scales

Singular Perturbations of Linear Ordinary Differential Equations .

Singular Perturbations of Second Order Elliptic Type. Linear Theory

Singular Perturbations of Second Order Hyperbolic Type.

Singular Perturbations in Nonlinear Initial Value Problems of Second Order

Singular Perturbations in Nonlinear Boundary Value Problems of Second Order

Perturbations of Higher Order

Theory of Orbits

Volume 1: Integrable Systems and Non-perturbative Methods

Introduction - The Theory of Orbits from Epicycles to "Chaos

Dynamics and Dynamical Systems - Quod Satis

The N-Body Problem

The Three-Body Problem

Orbits in Given Potentials

Mathematical Appendix

Topics in Black Hole Perturbation Theory and the Visualization of Curved Spacetime

Overview and Summary

Topics in Black Hole Perturbation Theory

New Generic Ringdown Frequencies at the Birth of a Kerr Black Hole

Quasinormal-Mode Spectrum of Kerr Black Holes and Its Geometric Interpreta- tion

Branching of Quasinormal Modes for Nearly Extremal Kerr Black Holes

Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime

Classifying the Isolated Zeros of Asymptotic Gravitational Radiation by Tendex and Vortex Line

Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes

Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition

Introduction

Overview and summary of contributions

Algorithms for matrix decompositions

Communication minimizing algorithm for the polar decomposition

Efficient, communication minimizing algorithm for the symmetric eigenvalue problem

Efficient, communication minimizing algorithm for the SVD

dqds with aggressive early deflation for computing singular values of bidiagonal matrices

Eigenvalue perturbation theory

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

Perturbation of generalized eigenvalues

Perturbation and condition numbers of a multiple generalized eigenvalue

Perturbation of eigenvectors

Gerschgorin theory

Overview and summary of contributions

Notations

The symmetric eigendecomposition and the singular value decomposition

The polar decomposition

Backward stability of an algorithm

The dqds algorithm

Aggressive early deflation

Hermitian eigenproblems

Generalized eigenvalue problems

Eigenvalue first-order expansion

Perturbation of eigenvectors

Gerschgorin’s theorem

Efficient, communication minimizing algorithm for the symmetric eigenvalue problem

Algorithm QDWH-eig

Practical considerations

Numerical experiments

Efficient, communication minimizing algorithm for the SVD

Algorithm QDWH-SVD

Practical considerations

Numerical experiments

dqds with aggressive early deflation for computing singular values of bidiagonal matrices

Aggressive early deflation for dqds - version 1: Aggdef(1)

Aggressive early deflation for dqds - version 2 Aggdef(2)

Convergence analysis

Numerical experiments

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

Basic approach

2-by-2 block case

Block tridiagonal case

Two case studies

Effect of the presence of multiple eigenvalues

Perturbation of generalized eigenvalues

Absolute Weyl theorem

Relative Weyl theorem

Quadratic perturbation bounds for Hermitian definite pairs

An extension to non-Hermitian pairs

Perturbation and condition numbers of a multiple generalized eigenvalue

Perturbation bounds for multiple generalized eigenvalues

Another explanation

Implication in the Rayleigh-Ritz process

Condition numbers of a multiple generalized eigenvalue

Hermitian definite pairs

Non-Hermitian pairs

Multiple singular value

Perturbation of eigenvectors

The tanθ theorem under relaxed conditions

The generalized tanθ theorem with relaxed conditions

Refined Rayleigh-Ritz approximation bound

New bounds for the angles between Ritz vectors and exact eigenvectors

Singular vectors

The cosθ theorem

Gerschgorin theory

A new Gerschgorin-type eigenvalue inclusion set

Examples and applications

Gerschgorin theorems for generalized eigenproblems

Examples

Applications

A Perturbation Theory for Hamilton’s Principal Function: Applications to Boundary Value Problems

Introduction

Hamiltonian Dynamics

Perturbation Theory for Hamilton’s Principal Function

Perturbation Theory for the Two-Point Boundary Value Problem

Perturbation Theory for the Initial Value Problem

Perturbed Rotating Two-Body Problem

Example: Solving the Perturbed Two-Body Two-Point Boundary Value Problem

Introduction

Celestial Mechanics and Mathematics

Two-Point Boundary Value Problems

Initial Value Problems

Perturbations

Hamiltonian Dynamics

Lagrangian Mechanics

Hamiltonian Dynamical Systems

Hamilton’s Principal Function

Hamiltonian Dynamical Systems

Hamilton’s Characteristic Function

Analogy between Principal, Characteristic, and Generating Functions

Phase Space

Example: Solving the Perturbed Two-Body Initial Value Problem

Principle of Least Action

Hamilton’s Equations

Hamilton’s Principle

Symplectic Structure of Hamiltonian Dynamical Systems

Generating Functions

Hamilton-Jacobi Equations

Perturbation Theory for Hamilton’s Principal Function

Perturbation Theory for Hamilton’s Characteristic Function

Perturbation Theory for the Two-Point Boundary Value Problem

Two-Point Boundary Value Problem Solutions for Hamilton’s Principal Function

Two-Point Boundary Value Problem Solutions for Hamilton’s Characteristic Function

Numerical Error Analysis of Perturbation Theory for The Two- Point Boundary Value Problem

Analytical Example: 1-Dimensional Particle Dynamics

Perturbation Theory for the Initial Value Problem

Perturbation Theory for the Initial Value Problem

First-Order Perturbation Theory

Perturbed Rotating Two-Body Problem

The Two-Body Problem

Keplerian Two-Body Problem

Delaunay Elements

Poincare Elements

The Restricted Three-Body Problem

The Perturbed Rotating Two-Body Problem

Orbit Transfers and Impulsive Maneuvers

Lambert’s Problem

Example: Solving the Perturbed Two-Body Two-Point Boundary Value Problem

Hamilton’s Principal Function for the Two-Body Problem

Perturbation Theory for the Two-Body Problem Hamilton’s Principal Function

Implementation of Perturbation Theory to Numerical Simulations

Perturbation Theory for the Hohmann Transfer

Perturbation Theory for Lambert’s Problem .

ContourMap for the Nominal Two-Point Boundary Value Problem

Implementation of Perturbation Theory to Perturbed Contour Map

Numerical Errors in LEO to GEO Transfer

Example: Perturbed LEO to GEO Hohmann Transfer

Physical Explanation of Perturbed Contour Map Results

Example: Solving the Perturbed Two-Body Two-Point Initial Value Problem

Perturbation Theory for the Two-Body Problem Hamilton’s Principal and Characteristic Functions

Implementation of the First-Order Perturbation Theory to Numerical Simulations

Numerical Example: Planar Perturbed Two-Body Problem

Application of the First-Order Perturbation Theory to the Perturbed Restricted Three-Body Problem

Interpretation of Numerical Results from Perturbation Theory

A Primer for Chiral Perturbation Theory

Quantum Chromodynamics and Chiral Symmetry

Spontaneous Symmetry Breaking and the Goldstone Theorem

Chiral Perturbation Theory for Mesons

Chiral Perturbation Theory for Baryons

Applications and Outlook

Appendix A: Pauli and Dirac Matrices

Appendix B: Functionals and Local Functional Derivatives

Quantum Chromodynamics and Chiral Symmetry

Some Remarks on SU(3)

Local Symmetries and the QCD Lagrangian

Accidental, Global Symmetries of the QCD Lagrangian

Green Functions and Ward Identities

Spontaneous Symmetry Breaking and the Goldstone Theorem

Degenerate Ground States

Spontaneous Breakdown of a Global, Continuous, Non-Abelian Symmetry

Goldstone Theorem

Explicit Symmetry Breaking: A First Look

Chiral Perturbation Theory for Mesons

Effective Field Theory

Spontaneous Symmetry Breaking in QCD

Transformation Properties of the Goldstone Bosons

Effective Lagrangian and Power-Counting Scheme

Beyond Leading Order

Chiral Perturbation Theory for Baryons

Transformation Properties of the Fields

Baryonic Effective Lagrangian at Lowest Order

Applications at Lowest Order

The Next-to-Leading-Order Lagrangian

Loop Diagrams: Renormalization and Power Counting

Renormalization Schemes.

The Delta Resonance

Applications and Outlook

Nucleon Mass and Sigma Term

Nucleon Electromagnetic Form Factors to O(q4)

Advanced Applications and Outlook

Nucleon Mass and Sigma Term

Nucleon Mass to O(q3) in the Heavy-Baryon Formalism

Nucleon Mass and Sigma Term at O(q4)

Nucleon Mass Including the Delta Resonance

Nucleon Mass to O(q6)

Local Symmetries and the QCD Lagrangian

The QED Lagrangian

The QCD Lagrangian

Accidental, Global Symmetries of the QCD Lagrangian.

Light and Heavy Quarks

Left-Handed and Right-Handed Quark Fields

Noether Theorem

Global Symmetry Currents of the Light-Quark Sector

The Chiral Algebra .

Chiral Symmetry Breaking by the Quark Masses

Green Functions and Ward Identities

Ward Identities Resulting from U(1) Invariance: An Example

Chiral Green Functions

The Algebra of Currents

QCD in the Presence of External Fields and the Generating Functional

PCAC in the Presence of an External Electromagnetic Field

Applications at Lowest Order

Goldberger-Treiman Relation and the Axial-Vector Current Matrix Element

Pion-Nucleon Scattering.

Loop Diagrams: Renormalization and Power Counting

Counter Terms of the Baryonic ChPT Lagrangian.

Power Counting

One-Loop Correction to the Nucleon Mass . .

Renormalization Schemes

Heavy-Baryon Approach

Infrared Regularization

Extended On-Mass-Shell Scheme

Dimensional Counting Analysis .

The Delta Resonance

The Free Lagrangian of a Spin-3/2 System

Isospin

Leading-Order Lagrangian of the D(1232) Resonance

Consistent Interactions

Spontaneous Symmetry Breaking in QCD

The Hadron Spectrum

The Scalar Singlet Quark Condensate

Transformation Properties of the Goldstone Bosons.

General Considerations

Application to QCD

Effective Lagrangian and Power-Counting Scheme

The Lowest-Order Effective Lagrangian

Construction of the Effective Lagrangian .

Application at Lowest Order: Pion Decay

Application at Lowest Order: Pion-Pion Scattering

Application at Lowest Order: Compton Scattering

Dimensional Regularization

The Generation of Counter Terms

Power-Counting Scheme

Effective Lagrangian and Power-Counting Scheme

The Lowest-Order Effective Lagrangian

Symmetry Breaking by the Quark Masses

Construction of the Effective Lagrangian .

Application at Lowest Order: Pion Decay

Application at Lowest Order: Pion-Pion Scattering

Application at Lowest Order: Compton Scattering

Dimensional Regularization

The Generation of Counter Terms.

Power-Counting Scheme

Goldberger-Treiman Relation and the Axial-Vector Current Matrix Element

Pion-Nucleon Scattering.

Advanced Applications and Outlook

Chiral Extrapolations

Pion Photo- and Electroproduction

Compton Scattering and Polarizabilities

Virtual Compton Scattering

Isospin-Symmetry Breaking

Three-Flavor Calculations

Chiral Unitary Approaches

Complex-Mass Scheme

Chiral Effective Theory for Two- and Few-Nucleon Systems

Exploring Perturbation Methods Together

We specialize in asymptotic approximations and solutions to differential equations, including Taylor's theorem, Kummer functions, and various problems in linear wave propagation and weakly nonlinear difference equations.

Our Expertise Awaits

Innovative Solutions Delivered

Join us as we delve into partial differential equations, elliptic and parabolic problems, and the discrete WKB method, providing insights and solutions for complex mathematical challenges.

Perturbation Methods Services

Specializing in asymptotic approximations and differential equations solutions for complex problems.

Differential Equations Solutions

Expert solutions for partial differential equations and elliptic or parabolic problems.

Taylor's Theorem Insights

Understanding Taylor's theorem and its applications in perturbation methods and approximations.

Wave Propagation Analysis

Analyzing linear wave propagation and weakly nonlinear difference equations for practical applications.

Mathematical Insights

Explore asymptotic methods, differential equations, and advanced mathematical concepts.

PERTURBATION

Introduction to Perturbation Methods

Introduction to Perturbation Techniques

Introduction to the Perturbation Theory of Hamiltonian Systems

Many-Body Perturbation Theory for Ab Initio Nuclear Structure

Mathematical Foundations of Quantum Field Theory and Perturbative String Theory

Nonlinear Singular Perturbation Phenomena: Theory and Applications

NON-PERTURBATIVE FIELD THEORY

From Two-Dimensional Conformal Field Theory to QCD in Four Dimensions

PART I NON-PERTURBATIVE METHODS IN TWO-DIMENSIONAL FIELD THEORY

PART II TWO-DIMENSIONAL NON-PERTURBATIVE GAUGE DYNAMICS

PART III FROM TWO TO FOUR DIMENSIONS

NON PERTURBATIVE QUANTUM FIELD THEORY

Mathematical Aspects and Applications

Perturbation Methods

Introduction

Straightforward Expansions and Sources of No n u n i fo r m i ty

The Method of Strained Coordinates

Variation of Parameters and Methods of Averaging

The Method of Multiple Scales

Asymptotic Solutions of Linear Equations

Examples and Motivations

Pertubation in Critical Point Theory

Bifurcation from the Essential Spectrum

Elliptic Problems on Rn with Subcritical Growth

Elliptic Problems with Critical Exponent

The Yamabe Problem

Other Problems in Conformal Geometry

Nonlinear Schr¨odinger Equations

Singularly Perturbed Neumann Problems

Concentration at Spheres for Radial Problems

Perturbation Methods for Differential Equations

Perturbation Methods, Bifurcation Theory and Computer Algebra

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations

Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

Perturbative Quantum Chromodynamics

Perturbative Quantum Electrodynamics and Axiomatic Field Theory

What Is Quantum Electrodynamics?

Perturbation Theory

Particles and Their Reactions

Singular Perturbation Methods for Ordinary Differential Equations

Examples Illustrating Regular and Singular Perturbation Concepts

Singularly Perturbed Initial Value Problems

Singularly Perturbed Boundary Value Problems

Singular Perturbation Methods in Control Analysis and Design

TIME-SCALE MODELING

LINEAR TIME-INVARIANT SYSTEMS

LINEAR FEEDBACK CONTROL

STOCHASTIC LINEAR FILTERING AND CONTROL

LINEAR TIME-VARYING SYSTEMS

OPTIMAL CONTROL

Adiabatic Perturbation Theory in Quantum Dynamics

Introduction

First order adiabatic theory

Time-dependent Born-Oppenheimer theory: Part I

Constrained quantum motion

Semiclassical limit for effective Hamiltonians

Space-adiabatic perturbation theory

Applications and extensions

Quantum dynamics in periodic media

Adiabatic perturbation theory for Bloch bands

Adiabatic decoupling without spectral gap

Effective N-body dynamics in the massless Nelson model

Pseudodifferential operators

Pseudodifferential operators

Asymptotic Methods and Perturbation Theory

PART I FUNDAMENTALS

PART II LOCAL ANALYSIS

PART III PERTURBATION METHODS

PART IV GLOBAL ANALYSIS

Ordinary Differential Equations

Difference Equations

Approximate Solution of Linear Differential Equations

Approximate Solution of Nonlinear Differential Equations

Approximate Solution of Difference Equations

Approximate Expansion of Integrals

Perturbation Series

Summation Series

Boundary Layer Theory

WKB Theory

Differential Equations

Asymptotic Series Solutions for Differential Equations

Difference Equations