Scattering Resonances of Microstructures and Homogenization Theory

Scattering resonances are the eigenvalues and corresponding eigenmodeswhich solve the Schrodinger equation $H\psi=E\psi$ for a Hamiltonian, H, subject to the condition of outgoing radiation at infinity. We consider the scattering resonance problem for potentials which are rapidly varying in space and are not necessarily small in a pointwise sense. Such potentials are of interest in many applications in quantum, electromagnetic, and acoustic scattering, where the environment consists of microstructure, e.g., rapidly varying material properties. Of particular interest in applications are high contrast microstructures, e.g., large pointwise variations of material properties.We develop a perturbation theory for the scattering resonances and eigenvalues of such high contrast and oscillatory potentials. The expansion is proved to be convergent in a norm which encodes the degree of oscillation in the microstructure. Next, we consider the concrete example of two-dimensional microstructure potentials. These corres...

[1]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[2]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[3]  Leon Poladian,et al.  Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry. , 2002, Optics express.

[4]  Teruo Ikebe,et al.  Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory , 1960 .

[5]  G. R. Hadley,et al.  Optical Waveguide Theory and Numerical Modelling , 2004 .

[6]  C. Kittel.  Introduction to solid state physics , 1954 .

[7]  Rustem R. Gadyl'shin,et al.  On an analog of the Helmholtz resonator in the averaging theory , 1999 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  Peter Kuchment,et al.  Waves in Periodic and Random Media , 2003 .

[10]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[11]  A. Stentz,et al.  Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm , 2000 .

[12]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[13]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[14]  J. Jasapara,et al.  Chromatic dispersion measurements in a photonic bandgap fiber , 2002, Optical Fiber Communication Conference and Exhibit.

[15]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[16]  Melrose,et al.  Geometric Scattering Theory , 1995 .

[17]  Рустем Рашитович Гадыльшин,et al.  Существование и асимптотики полюсов с малой мнимой частью для резонатора Гельмгольца@@@Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator , 1997 .

[18]  Jelena Vucković,et al.  Design of photonic crystal microcavities for cavity QED. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  W. Donoghue.  Distributions and Fourier transforms , 1969 .

[20]  S. Moskow,et al.  First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  G. Stewart.  Optical Waveguide Theory , 1983, Handbook of Laser Technology and Applications.

[22]  D. Marcuse.  Theory of dielectric optical waveguides , 1974 .

[23]  Michael Vogelius,et al.  First-Order Corrections to the Homogenized Eigenvalues of a Periodic Composite Medium , 1993, SIAM J. Appl. Math..

[24]  Steven E. Golowich,et al.  Homogenization expansion for resonances of microstructured photonic waveguides , 2003 .

[25]  Norman Shenk,et al.  Resonant states and poles of the scattering matrix for perturbations of −Δ , 1972 .

[26]  J. B. McLeod,et al.  THE ANALYTIC CONTINUATION OF THE RESOLVENT KERNEL AND SCATTERING OPERATOR ASSOCIATED WITH THE SCHROEDINGER OPERATOR. , 1966 .