Alexander Figotin

Spectra of random and almost-periodic operators

I. Metrically Transitive Operators.- 1 Basic Definitions and Examples.- 1.A Random Variables, Functions and Fields.- 1.B Random Vectors and Operators.- l.C Metrically Transitive Random Fields.- l.D Metrically Transitive Operators.- 2 Simple Spectral Properties of Metrically Transitive Operators.- 2.A Deficiency Indices.- 2.B Nonrandomnessofthe Spectrum and of its Components.- 2.C Nonrandomness of Multiplicities.- Problems.- II. Asymptotic Properties of Metrically Transitive Matrix and Differential Operators.- 3 Review of Basic Results.- 4 Matrix Operators on ?2 (Zd).- 4.A Essential Self-Adjointness.- 4.B Existence of the Integrated Density of States and Other Ergodic Properties.- 4.C Simple Properties of the Integrated Density of States and of the Spectra of Metrically Transitive Matrix Operators.- 4.D Location of the Spectrum.- 5 Schrodinger Operators and Elliptic Differential Operators on L2(Rd).- 5.A Criteria for Essential Self-Adjointness.- 5.B Ergodic Properties.- 5.C Some Properties of the Integrated Density of States.- 5.D Location of the Spectrum of a Metrically Transitive Schrodinger Operator.- Problems.- III. Integrated Density of States in One-Dimensional Problems of Second Order.- 6 The Oscillation Theorem and the Integrated Density of States.- 6. A The Phase and the Existence of the Integrated Density of States.- 6.B Simplest Asymptotics of the Integrated Density of States at the Edges of the Spectrum.- 6.C Schrodinger Operator with Markov Potential.- 6.D The Brownian Motion Model.- 6.E Jacobi Matrices with Independent and Markov Coefficients.- 6.F Smoothness of N (?) Special Energies.- 7 Examples of Calculation of the Integrated Density of States.- 7.A The Kronig-Penny Stochastic Model.- 7.B Random Jacobi Matrices.- Problems.- IV. Asymptotic Behavior of the Integrated Density of States at Spectral Boundaries in Multidimensional Problems.- 8 Stable Boundaries.- 9 Fluctuation Boundaries: General Discussion and Classical Asymptotics.- 9.A Introduction and Heuristic Discussion.- 9.B Simplest Bounds. Gaussian and Negative Poisson Potentials.- 9.C Generalized Poisson Potential.- 10 Fluctuation Boundaries: Quantum Asymptotics.- 10.A The Lifshitz Exponent.- 10.B Generalized Poisson Potential with a Nonnegative, Rapidly Decreasing Function.- 10.C Smoothed Square of a Gaussian Random Field.- Problems.- V. Lyapunov Exponents and the Spectrum in One Dimension.- 11 Existence and Properties of Lyapunov Exponents.- 11.A The Multiplicative Ergodic Theorem and the Existence of Lyapunov Exponents.- 11.B The Lyapunov Exponent and the Integrated Density of States.- 11.C Simplest Asymptotic Formulas and Estimates for Lyapunov Exponents.- 12 Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.A Basic Facts About the Spectrum of One-Dimensional Operators of the Second Order.- 12.B Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.C Multiplicity of the Spectrum.- 12.D Deterministic Potentials.- 12.E Some Inverse Problems.- 13 Lyapunov Exponents and the Point Spectrum.- 13.A Heuristic Discussion.- 13.B Conditions for Positive Lyapunov Exponents to Imply a Pure Point Spectrum.- Problems.- VI. Random Operators.- 14 The Lyapunov Exponent of Random Operators in One Dimension.- 14.A Positiveness of the Lyapunov Exponent.- 14.B Asymptotic Formulas for the Lyapunov Exponent.- 15 The Point Spectrum of Random Operators.- 15.A The Pure Point Spectrum in One Dimension.- 15.B Other One-Dimensional Results.- 15.C The Point Spectrum in Multidimensional Problems.- Problems.- VII. Almost-Periodic Operators.- 16 Smooth Quasi-Periodic Potentials.- 16.A The Integrated Density of States and the Gap Labeling Theorem.- 16.B Absolutely Continuous Spectrum.- 16.C Lower Bounds of Solutions and Absence of a Point Spectrum.- 16.D Lower Bounds for the Lyapunov Exponent and Absence of an Absolutely Continuous Spectrum in the Discrete Case.- 16.E Point Spectrum of Almost-Periodic Operators.- 16.F The Almost-Mathieu Operator.- 17 Limit-Periodic Potentials.- 17.A Basic Results.- 17.B Spectral Data for Periodic Potentials of Increasing Period.- 17.C Proof of the Main Theorems.- 18 Unbounded Quasiperiodic Potentials.- 18.A General Results and the Integrated Density of States.- 18.B The Case of Strongly Incommensurate Frequencies.- 18.C The One-Dimensional Case.- 18.D The Schrodinger Operator with a Nonlocal Quasiperiodic Potential.- Problems.- Appendix A: Nevanlinna Functions.- Appendix B: Distribution of Eigenvalues of Large Random Matrices.- List of Symbols.

Electromagnetic Unidirectionality in Magnetic Photonic Crystals

Magnetization, either spontaneous or field-induced, is always associated with nonreciprocal circular birefringence which breaks the reciprocity principle and qualitatively changes electrodynamics of medium. In magnetic photonic crystals and other periodic structures involving magnetic components, broken reciprocity can result in electromagnetic unidirectionality, when the traveling waves can only propagate in one the two opposite directions. The unidirectional wave propagation can only occur if both time reversal and space inversion symmetries of the periodic structure are broken. During the last decade there have been numerous publications devoted to this kind of phenomenon. Our goal is to present some of those ideas.

Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers

We consider Fabry-Perot cavity resonance in periodic stacks of anisotropic layers with misaligned in-plane anisotropy at the frequency close to a photonic band edge. We show that in-plane dielectric anisotropy can result in a dramatic increase in field intensity and group delay associated with the transmission resonance. The field enhancement turns out to be proportional to fourth degree of the number N of layers in the stack. By contrast, in common periodic stacks of isotropic layers, those effects are much weaker and proportional to N2 Thus, the anisotropy allows one to drastically reduce the size of the resonance cavity with similar performance. The key characteristic of the periodic arrays with gigantic transmission resonance is that the dispersion curve omega(k) at the photonic band edge has the degenerate form Deltaomega approximately (Deltak)4, rather than the regular form Deltaomega approximatley (Deltak)2. This can be realized in specially arranged stacks of misaligned anisotropic layers. The degenerate band-edge cavity resonance with similar outstanding properties can also be realized in a waveguide environment, as well as in a linear array of coupled multimode resonators, provided that certain symmetry conditions are in place.

Band-gap structure of spectra of periodic dielectric and acoustic median I: scalar model

We investigate the band-gap structure of the spectrum of second-order partial differential operators associated with the propagation of waves in a periodic two-component medium. The medium is characterized by a real-valued position-dependent periodic function

LOCALIZATION OF CLASSICAL WAVES. I: ACOUSTIC WAVES

\varepsilon ( x )

Band-gap structure of spectra of periodic dielectric and acoustic media II: two-dimensional photonic crystals

that is the dielectric constant for electromagnetic waves and mass density for acoustic waves. The imbedded component consists of a periodic lattice of cubes where

Spectral properties of classical waves in high-contrast periodic media

\varepsilon ( x ) = 1

Localized Classical Waves Created by Defects

. The value of

Slow light in photonic crystals

\varepsilon ( x )

Midgap defect modes in dielectric and acoustic media

on the background is assumed to be greater than 1. We give the complete proof of existence of gaps in the spectra of the corresponding operators provided some simple conditions imposed on the parameters of the medium.