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

    References

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  • 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