The Weyl Operator and its generalization Leon Cohen

Format:

Book

Author/Creator:

Cohen, Leon, 1940-

Series:

Pseudo-differential operators, theory and applications v. 9

Language:

English

Subjects (All):

Generalized spaces.

Mathematical physics.

Physical Description:

1 online resource

Place of Publication:

Basel New York Springer 2013

Language Note:

English

System Details:

text file

PDF

Summary:

This book deals with the theory and application of associating a function of two variables with a function of two operators that do not commute. The concept of associating ordinary functions with operators has arisen in many areas of science and mathematics, and up to the beginning of the twentieth century many isolated results were obtained. These developments were mostly based on associating a function of one variable with one operator, the operator generally being the differentiation operator. With the discovery of quantum mechanics in the years 1925-1930, there arose, in a natural way, the issue that one has to associate a function of two variables with a function of two operators that do not commute. Methods to do so became known as rules of association, correspondence rules, or ordering rules. This has led to a wonderfully rich mathematical development that has found applications in many fields. Subsequently it was realized that for every correspondence rule there is a corresponding phase-space distribution. Now the fields of correspondence rules and phase-space distributions are intimately connected. A similar development occurred in the field of time-frequency analysis where the aim is to understand signals with changing frequencies. The Weyl Operator and Its Generalization aims at bringing together the basic results of the field in a unified manner. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner

Contents:

Introduction and Terminology Operator Algebra The Weyl Operator Generalized Operator Association Generalized Phase-Space Distributions Special Cases Unitary Transformation Path Integral Approach Time-Frequency Operators Transformation of Differential Equations Into Phase Space The Eigenvalue Problem in Phase-Space Arbitrary Operators: Single Operator Uncertainty Principle for Arbitrary Operators The Khintchine Theorem and Characteristic Function Representability Arbitrary operators: Two Operators

Introduction

The Fundamental Idea, Terminology, and Operator Algebra

The Weyl Operator

The Algebra of the Weyl Operator

Product of Operators, Commutators, and the Moyal Sin Bracket

Some Other Ordering Rules

Generalized Operator Association

The Fourier, Monomial, and Delta Function Associations

Transformation Between Associations

Path Integral Approach

The Distribution of a Symbol and Operator

The Uncertainty Principle

Phase-Space Distributions

Amplitude, Phase, Instantaneous Frequency, and the Hilbert Transform

Time

Frequency Analysis

The Transformation of Differential Equations into Phase Space

The Representation of Functions

The N Operator Case

Notes:

Includes bibliographical references and index

Print version record

Other Format:

Print version Cohen, Leon. Weyl Operator and its generalization

ISBN:

9783034802949

3034802943

9781283934626

1283934620

OCLC:

822976870

Access Restriction:

Restricted for use by site license