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PERIODS OF EISENSTEIN AND POINCARÉ SERIES

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0010
Subject:
Mathematics, Mathematical Physics

This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in ... More


Basic Quantum Mechanics

Stephen C. Rand

in Lectures on Light: Nonlinear and Quantum Optics using the Density Matrix

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199574872
eISBN:
9780191722219
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199574872.003.0002
Subject:
Physics, Atomic, Laser, and Optical Physics

Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave‐like behavior is an important part of the ... More


Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.001.0001
Subject:
Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More


The rotating wave van del Pol oscillator (RWVP)

Melvin Lax, Wei Cai, and Min Xu

in Random Processes in Physics and Finance

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198567769
eISBN:
9780191718359
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567769.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explains why a laser line is so narrow (it can be as low as one cycle per second, even when the laser frequency is of the order of 10 per second) and shows that autonomous oscillators ... More


Dirac Notation and Transformation Theory

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Published in print:
2017
Published Online:
January 2018
ISBN:
9780198791942
eISBN:
9780191834165
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198791942.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also ... More


Operators in quantum mechanics

Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski, Jr.

in Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199232710
eISBN:
9780191774973
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232710.003.0001
Subject:
Physics, Atomic, Laser, and Optical Physics

Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by ... More


On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

Christopher D. Sogge and Steve Zelditch

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159416
eISBN:
9781400848935
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159416.003.0018
Subject:
Mathematics, Numerical Analysis

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a ... More


A review: The Laplacian and the d’Alembertian

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0001
Subject:
Mathematics, Numerical Analysis

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be ... More


Classical and quantum ergodicity

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0006
Subject:
Mathematics, Numerical Analysis

This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the ... More


Basic Quantum Mechanics

Stephen C. Rand

in Lectures on Light: Nonlinear and Quantum Optics using the Density Matrix

Published in print:
2016
Published Online:
August 2016
ISBN:
9780198757450
eISBN:
9780191817830
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198757450.003.0002
Subject:
Physics, Atomic, Laser, and Optical Physics

Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the ... More


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