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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

## Periodic Hamiltonians and the Emergence of Band Structure: The Bloch Theorem and the Dirac Kronig–Penney Model

*Duncan G. Steel*

### in Introduction to Quantum Nanotechnology: A Problem Focused Approach

- Published in print:
- 2021
- Published Online:
- July 2021
- ISBN:
- 9780192895073
- eISBN:
- 9780191924149
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192895073.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Crystals are defined by their atomic makeup and their crystal symmetry. The spatial symmetry leads to a periodic potential that localizes the electron or carrier at the lattice sites corresponding to ... 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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