Quantization Methods in the Theory of Differential Equations - Volume 3

V.E. Nazaikinskii, B.W. Schulze, B.Yu. Sternin

356 pages, parution le 02/07/2002

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Résumé

This volume:

gives a systematic study of quantization-based methods for linear partial differential equations
describes quantization of all types of classical objects (states, observables, canonical transformations)
considers semiclassical asymptotic quantization based on the wave packet transform and quantization of Poisson brackets via noncommutative analysis
provides unified treatment of various kinds of asymptotics
includes numerous examples helping the reader understand the material
outlines a wide range of applications from differential equations to theoretical physics
examines propagation of electromagnetic waves in plasma and ionosphere channelling of high frequency signals
constructs asymptotic and approximate solutions of 3D Maxwell equations

The volume is intended for scientists, undergraduates and graduate students specializing in differential equations, applied mathematics and mathematical and theoretical physics.

Contents

I Semiclassical Quantization
1 Quantization and Microlocalization
- Classical mechanics and quantum mechanics
- The correspondence principle and the quantization problem
- Coordinate and momentum representations and the Fourier transform
- Hamiltonians
- Semiclassical states
- The oscillation front and the problem of microlocal analysis
2 Quantization by the Wave Packet Transform
- Quantization of pure states
- The wave packet transform
- Quantization of observables
- Quantization of mixed states
- Quantization of canonical transformations
- Quantization in the wave packet representation
3 Maslov's Canonical Operator and Hormander's Oscillatory Integrals
- The classification problem for oscillatory integrals
- Transformations of phase functions
- The local classification of nondegenerate phase functions; Normal forms
- The classification of oscillatory integrals and the canonical operator
- Simultaneous asymptotics
4 Topological Aspects of Quantization Conditions
- Cech cocycles and obstructions to the existence of the canonical operator
- Quantized measures and a differential form representing the Maslov-Arnold class
- The index of paths on Lagrangian manifolds
- Quantization of symplectic manifolds
5 The Schrodinger Equation
- The Cauchy problem
- The spectrum asymptotics
- Asymptotics of the Green function
6 The Maxwell Equations
- Statement of the problem
- Solution of the homogeneous Maxwell equations
- Asymptotics of the Green function
- Solution of the nonhomogeneous Maxwell equations
- The Helmholtz equation on the sphere
7 Equations with Trapping Hamiltonians
- Statement of the problem
- A scheme for constructing the regularizer
- The contribution of nonclosed trajectories
- The contribution of closed trajectories
- Example: propagation of short electromagnetic waves in ionosphere

II Quantization by the Method of Ordered Operators (Noncommutative Analysis)
8 Noncommutative Analysis: Main Ideas, Definitions, and Theorems
- Functions of one operator
- Functions of several operators
- Main formulas of operator calculus
- Main tools of noncommutative analysis
- Composition laws and ordered representations
9 Exactly Soluble Commutation Relations
- Statement of the problem
- Some examples
- Lie commutation relations
- Nonlinear commutation relations
10 Operator Algebras on Singular Manifolds
- Statement of the problem
- Operators on the model cone
- Operators on the model cusp of order A:
- Application: the construction of regularizers and proof of the finiteness theorem
11 The High-Frequency Asymptotics in the Problem of Wave Propagation in Plasma
- Statement of the problem
- Mixed asymptotics: the general scheme
- The asymptotic solution of the main problem
- Analysis of the asymptotic solution

Appendices
A Complex Pseudocoordinates and Classical Mechanics
- Complex pseudocoordinates
- Classical mechanics in complex pseudocoordinates
B Some Formulas Related to the Wave Packet TVansform and Quantization

Caractéristiques techniques

	PAPIER
Éditeur(s)	Taylor and Francis Books
Auteur(s)	V.E. Nazaikinskii, B.W. Schulze, B.Yu. Sternin
Parution	02/07/2002
Nb. de pages	356
Format	17,5 x 25,5
Couverture	Relié
Poids	800g
Intérieur	Noir et Blanc
EAN13	9780415273640
ISBN13	978-0-415-27364-0