Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy

2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.We give a complete pointwise asymptotic expansion for the Spectral Shift Function for Schrödinger operators that are perturbations of the Laplacian on Rn with slowly decaying potentials

Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

1991 Mathematics Subject Classification. Primary 37K55, 35B10, 35J10, 35Q40, 35Q55.In this paper, one-dimensional (1D) nonlinear Schrödinger equation [equation omitted] with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small-amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkho ff normal form reduction and an improved KAM method.Partially supported by NSF grant DMS0204119

A recipe for making materials with negative refraction in acoustics

A recipe is given for making materials with negative refraction in acoustics, i.e., materials in which the group velocity is directed opposite to the phase velocity. The recipe consists of injecting many small particles into a bounded domain, filled with a material whose refraction coefficient is known. The number of small particles to be injected per unit volume around any point $x$ is calculated as well as the boundary impedances of the embedded particles

Many-body wave scattering by small bodies

Scattering problem by several bodies, small in comparison with the wavelength, is reduced to linear algebraic systems of equations, in contrast to the usual reduction to some integral equations

A method for creating materials with a desired refraction coefficient

It is proposed to create materials with a desired refraction coefficient in a bounded domain $D\subset \R^3$ by embedding many small balls with constant refraction coefficients into a given material. The number of small balls per unit volume around every point $x\in D$, i.e., their density distribution, is calculated, as well as the constant refraction coefficients in these balls. Embedding into $D$ small balls with these refraction coefficients according to the calculated density distribution creates in $D$ a material with a desired refraction coefficient