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Spectrum of Differential Operators in Modern Electromagnetics - MATH 8970
This course is taught jointly by Profs. Stephen Shipman (Louisiana State University) and Junshan Lin (Auburn University) in a live-streaming setting
via Zoom . Students from any SEC university may enroll.
    Course Topics
- Chapter 1. Layer potential and boundary integral equations
- 1.1 Sobolve spaces
- 1.2 Layer potential theory for Laplace equation
- 1.3 Neumann-Poincare operator
- 1.4 Layer poential and boundary integral equations for Helmholtz equations
- 1.5 Numerical disretizations of singular integral operators
- 1.6 Periodic Green's functions and their computations
- Chapter 2. Plasmon for nano-particles
- 2.1 Boundary inegral method for plasmon of nano-particles
- 2.2 Analytical approaches for plasmon of nano-paticles
- 2.3 General properties for the spectrum of Neumann-Poincare operator
- 2.4 Plasmon for nano-particles with corners
- Chapter 3. Surface plasmon polariton
- 3.1 Surface plasmon modes and dispersion curve
- 3.2 Excitation of surface plamson polariton
- Chapter 4. Extraordiary optical transmission (EOT) through nano-holes
- 4.1 Resonance for a single hole
- 4.2 Resonance for a periodic array of holes; embedded eigenvalues
- 4.3 EOT for plamsonic metals with small holes
    Lecture Notes
- Jan 10
- Jan 15, Jan 17, Jan 22
- Jan 24, Jan 29
- Jan 29, Jan 31, Feb 5
- Feb 5, Feb 7, Feb 12
- Feb 14, Feb 19, Feb 21
- Feb 26, Feb 28, March 7
- March 19, March 21
- March 21 , March 26 , March 28
- April 2
- April 23
    References
- R. Adams and J. Fourier, Sobolev Spaces, Aacademic Press, 2013
- H. Ammari, et al., Mathematical and Computational Methods in Photonics and Phononics, Americal Mathematical Society, 2018
- H. Ammari, et al., Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Archive for Rational Mechanics and Analysis, Vol 208, 667-692, 2013. [PDF]
- E. Bonnetier and H. Zhang, Characterization of the essential spectrum of the Neumann-Poincar\'e operator in 2D domains with corner via Weyl sequences [PDF]
- D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, SIAM, 2013
- G. Hsiao and W. Wendland, Boundary Integral Equations, Springer, 2008
- J. Helsing, Solving Integral Equations on Piecewise Smooth Boundaries Using the RCIP Method: A Tutorial
[PDF]
- R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, JCAM, 1995, 345-360
[PDF]
- R. Kress, Linear Integral Equations, 3rd edition, Springer, 2014
- S. Maier, Plasmonics: Fudamentals and Applications, Springer 2007
- I. Mayergoyz, Plasmon Resonances in Nanoparticles, World Scientific Publisher, 2013
- Y. Miyanishi and T. Suzuki,Eigenvalues and eigenfunctions of double layer potentials, Transactions of the American Mathematical Society, 2017, 8037-8059. [PDF]
- K. Perfekt and M. Putinar, The essential spectrum of the Neumann–Poincaré operator on a domain with corners, Archive for Rational Mechanics and Analysis, 2017, 1019-1033. [PDF]
- E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SINUM, 1986, 1-10. [PDF]
- G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1995.
- S. Zoalroshd, On Spectral Properties of Single Layer Potentials, 2016
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