Contents
1 Introduction and Preliminaries 1
1.1 Overview 1
1.2 Layer Potentials in Electro-Magnetic System 2
13 Layer Potentials in Elastic System 4
1.4 Bessel and Neumann Functions 6
2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime 9
2.1 Maxwell's Problem 9
2.1.1 Introduction to Plasmonic Resonances 9
2.1.2 Drude's Model for the Electric Permittivity and Magnetic Permeability 12
2.1.3 Boundary Integral Operators and Resolvent Estimates 15
2.1.4 Layer Potential Formulation 24
2.1.5 Derivation of the Asymptotic Formula 25
2.1.6 Numerical Ilustrations 46
2.1.7 Concluding Remarks 48
2.2 Elastic Problem 48
2.2.1 Layer Potential Techniques 51
2.2.2 Asymptotics for the Integral Operators 53
2.23 Far Field Expansion 57
2.2.5 Resolvent Analysis 67
2.2.6 Polariton Resonance for Elastic Nanoparticles 69
3 Anomalous Localized Resonances and Their Cloaking Effect 77
3.1 Elastostatic Problem 77
3.1.1 Mathematical Setup of Elastostatics Problem 77
3.1.2 Preliminaries on Layer Potentials 79
3.1.3 Spectral Analysis of N-P Operator in Spherical Geometry 81
3.1.4 Anomalous Localized Resonances and Their Cloaking Effect 90
3.1.5 Cloaking by Anomalous Localized Resonance on a Coated Structure in Two Dimensional Case 97
3.2 Electrostatic Problem 115
3.2.1 Background 115
3.2.2 Layer Potential Formulation and Spectral Theory of a Neumann Poincare -Type Operator 117
3.2.3 Analysis of Cloaking Due to Anomalous Localized Resonance 121
4 Localized Resonances for Anisotropic Geometry 129
4.1 Conductivity Problem 129
4.1.1 Some Auxiliary Results 130
4.1.2 Quantitative Analysis of the Electric Field 135
4.13 Application to Calderon's Inverse Inclusion Problem 144
4.2 Helmholtz Problem 145
4.2.1 Asymptotic and Quantitative Analysis of the Scattering Field 150
4.2.2 Resonance Analysis of the Exterior Wave Field 169
4.2.3 Resonance Analysis of the Interior Wave Field 176
4.2.4 Conclusion 181
5 Localized Resonances Beyond the Quasi-Static Approximation 183
5.1 Spectral System of Neumann Poincare Operators in Helmoholtz System and Its Asymptotic Behavior 184
5.1.1 Layer Potential and Spectral Properties of Neumann Poincare Operator in R3 184
5.1.2 Asymptotic Behavior of Spectral System of Ncumann-Poincare Operator 187
5.1.3 Two Dimensional Case 191
5.2 Helmboltz System 196
5.2.1 Atypical Resonance and ALR Results in Three Dimensions 200
5.2.2 Spectral System of the N-P Operalor and IIs Application to Atypical Resonance in R3 210
5.2.3 Atypical Resonance and ALR Results in Two Dimensions 213
5.3 Maxwell's Problem 219
5.3.1 Integral Formulation of the Maxwell System 220
5.3.2 Spectral Analysis of the Integral Operators 225
5.3.3 Atypical Resonance and Its Cloaking Effect 233
5.3.4 Invisibility Cloaking 240
5.4 Elastic Problem 248
5.4.1 Preliminaries 251
5.4.2 Spectrum System of the Neumann-Poincare Operator 257
5.4.3 Atypical Resonance Beyond the Quasi -Static 267
5.4.4 CALR Beyond the Quasi-Static Approximation 273
6 Interior Transmission Resonance 281
6.1 Introduction 281
6.2 Scalar Case (Helmholtz Equations) 285
6.2.1 Boundary-Localized Transmission Eigenstates 285
6.2.2 Super-Resolution Wave Imaing 311
6.2.3 Numerical Example 321
6.2.4 Pseudo Surface Plasmon Resonances and Potential 324
6.2.5 Concluding Remarks and Discussions 325
6.3 Vectorial Case (Maxwell Equations) 328
6.3.1 Background 328
6.3.2 Boundary-L ocalized Transmission Eigenmodes 329
6.3.3 Numerics 348
6.3.4 Application of Boundary-Localized Transmission Eigenfunctions: Artificial Mirage 351
6.4 Concluding Remarks 356
References 357
