This monograph explains the theory of quantum waveguides, that is, dynamics of quantum particles confined to regions in the form of tubes, layers, networks, etc. The focus is on relations between the confinement geometry on the one hand and the spectral and scattering properties of the corresponding quantum Hamiltonians on the other. Perturbations of such operators, in particular, by external fields are also considered. The volume provides a unique summary of twenty-five years of research activity in this area and indicates ways in which the theory can develop further. The book is fairly self-contained. While it requires some broader mathematical physics background, all the basic concepts are properly explained and proofs of most theorems are given in detail, so there is no need for additional sources. Without a parallel in the literature, the monograph by Exner and Kovarik guides the reader through this new and exciting field.
Author(s): Pavel Exner; Hynek Kovařík
Series: Theoretical and Mathematical Physics
Publisher: Springer
Year: 2015
Cover
S Title
Theoretical and Mathematical Physics
Quantum Waveguides
© Springer International Publishing Switzerland 2015
ISSN 1864-5879
ISSN 1864-5887 (electronic)
ISBN 978-3-319-18575-0
ISBN 978-3-319-18576-7 (eBook)
DOI 10.1007/978-3-319-18576-7
Library of Congress Control Number: 2015938752
Dedication
Preface
Contents
Symbols
Introduction
1 Geometrically Induced Bound States
1.1 Smoothly Bent Strips
1.2 Polygonal Ducts
1.3 Bent Tubes in mathbbR3
1.4 Local Perturbations of Straight Tubes
1.5 Coupled Two-Dimensional Waveguides
1.5.1 A Lateral Window Coupling
1.5.2 A Leaky Interface
1.5.3 Crossed Strips
1.6 Thin Bent Tubes
1.7 Twisted Tubes
1.7.1 A Hardy Inequality for Twisted Tubes
1.7.2 Stability of the Spectrum
1.7.3 Periodically Twisted Tubes and Their Perturbations
1.8 Notes
1.9 Problems
2 Transport in Locally Perturbed Tubes
2.1 Existence and Completeness
2.2 The On-Shell S-Matrix: An Example
2.3 Resonances from Perturbed Symmetry
2.4 Resonances in Thin Bent Strips
2.5 Notes
2.6 Problems
3 More About the Waveguide Spectra
3.1 Spectral Estimates
3.1.1 Simple Bounds
3.1.2 Lieb-Thirring Inequalities
3.1.3 The Number of Eigenvalues in Twisted Waveguides
3.2 Related Results
3.2.1 Combined Boundary Conditions
3.2.2 Robin Boundary Conditions
3.2.3 An Isoperimetric Problem
3.2.4 Higher Dimensions
3.3 Interacting Particles
3.4 Acoustic Waveguides
3.4.1 Eigenvalues of the Neumann Laplacian in Tubes
3.4.2 Resonances in Acoustic Waveguides
3.5 Notes
3.6 Problems
4 Dirichlet Layers
4.1 Layers of Non-positive Curvature
4.1.1 Geometric Preliminaries
4.1.2 Curvature-Induced Bound States
4.2 More General Curved Layers
4.2.1 Other Sufficient Conditions
4.2.2 Layers with a Cylindrical Symmetry
4.3 Locally Perturbed Layers
4.4 Laterally Coupled Layers
4.5 Notes
4.6 Problems
5 Point Perturbations
5.1 Point Impurities in a Straight Strip
5.1.1 A Single Perturbation
5.1.2 A Finite Number of Impurities
5.2 Point Perturbations in a Tube
5.3 Point Perturbations in a Layer
5.4 Notes
5.5 Problems
6 Weakly Coupled Bound States
6.1 Birman-Schwinger Analysis
6.2 Applications to Tubes and Layers
6.2.1 Mildly Bent Tubes
6.2.2 Gently Curved Layers
6.2.3 A Direct Estimate: Local Deformations
6.3 A Generalized BS Technique
6.3.1 A Resolvent Formula
6.3.2 A Semitransparent Barrier
6.4 Variational Estimates
6.4.1 A Critically Deformed Strip
6.4.2 Window-Coupled Strips
6.4.3 Window-Coupled Layers
6.5 Distant Perturbations: Matching Methods
6.6 Notes
6.7 Problems
7 External Fields and Magnetic Transport
7.1 External Fields
7.1.1 Homogeneous Electric Fields
7.1.2 Local Magnetic Fields
7.1.3 Nöckel's Model Revisited
7.2 Magnetic Transport in Electron Gas
7.2.1 Edge States
7.2.2 Edge States Without a Classical Analogue
7.2.3 The Iwatsuka Model
7.3 Notes
7.4 Problems
8 Graph Limits of Thin Network Systems
8.1 Quantum Graphs
8.2 Vertex Coupling Approximations
8.2.1 δ-Coupling
8.2.2 δs'-Coupling
8.2.3 General Singular Vertex Coupling
8.3 An Abstract Convergence Result
8.3.1 Scale of Hilbert Spaces
8.3.2 Resolvent Convergence and Functional Calculus
8.3.3 Spectral Convergence
8.4 The Squeezing Limit of Neumann Networks
8.4.1 The Problem Setting
8.4.2 Spectral Convergence: Kirchhoff Coupling
8.4.3 Spectral Convergence: More General Couplings
8.5 The Squeezing Limit of Dirichlet Networks
8.6 Notes
8.7 Problems
9 Periodic and Random Systems
9.1 Periodic Waveguides
9.1.1 Absolute Continuity
9.1.2 Periodically Curved Waveguides
9.2 Periodic Point Perturbations
9.2.1 Point Perturbations in a Strip
9.2.2 Magnetic Layers with Periodic Point Perturbations
9.3 Random Waveguides
9.4 Notes
9.5 Problems
10 Leaky Waveguides
10.1 Leaky Graph Hamiltonians
10.2 Geometrically Induced Properties
10.2.1 Effects of Curvature and Local Deformations
10.2.2 Hiatus Perturbations
10.2.3 Isoperimetric Problem
10.3 Strong Coupling Asymptotics
10.3.1 Interactions Supported by Curves
10.3.2 Interactions Supported by Surfaces
10.3.3 Periodic and Magnetic Systems
10.4 Notes
10.5 Problems
Appendix A
Coda
Bibliography
Index
