Evgeny Lakshtanov

ELLIPTICITY IN THE INTERIOR TRANSMISSION PROBLEM IN ANISOTROPIC MEDIA

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) − 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

Remarks on interior transmission eigenvalues, Weyl formula and branching billiards

This paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.

Bounds on positive interior transmission eigenvalues

This paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.

A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy

We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or generic versions of the key results of this approach for the case of positive energy and compactly supported potentials. In particular, we do not assume that the potential is small or that Faddeev scattering solutions do not have singularities (i.e. we allow the Faddeev exceptional points to exist). Applications of these results to the Novikov-Veselov equation are also considered.

Total cross section exceeds transport cross section for quantum scattering from hard bodies at low and high wave numbers

The quantum scattering by smooth bodies is considered for small and large values of kd, with k the wave number and d the scale of the body. In both regimes, we prove that the forward scattering exceeds the backscattering. For high k, we need to assume that the body is strictly convex.

A Priori Estimates for High Frequency Scattering by Obstacles of Arbitrary Shape

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number k → ∞. This bound of the total cross section is sharp.

Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues

We consider the interior transmission eigenvalue (ITE) problem, which arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (when the latter approach {\bf

High Frequency Scattering by a Classically Invisible Body

z=1

Resonance regimes of scattering by small bodies with impedance boundary conditions

)}. We obtain a Weyl type formula for the counting function of positive ITEs, which are taken together with ascribed signs.