arXiv subjects

Alexei Iantchenko

Publications and source records attributed to Alexei Iantchenko.

23 records · Page 2Linked to original sources

arXiv2010

Resonance spectrum for one-dimensional layered media

We consider the "weighted" operator $P_k=-\partial_x a(x)\partial_x$ on the line with a step-like coefficient which appears when propagation of waves thorough a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of $P_k.$ If the coefficient is periodic on a finite interval (locally periodic) with $k$ identical cells then the resonance spectrum of $P_k$ has band structure. In the present paper we study a transition to semi-infinite medium by taking the limit $k\to \infty.$ The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem ($k=\infty$) with $k-1$ or $k$ resonances in each band. We prove that as $k\to \infty$ the resonance spectrum converges to the real axis.

math.SP↗

arXiv2010

Schrödinger operator on the zigzag half-nanotube in magnetic field

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schrödinger operator with a periodic potential plus a finitely supported perturbation. We describe all eigenvalues and resonances of this operator, and theirs dependence on the magnetic field. The proof is reduced to the analysis of the periodic Jacobi operators on the half-line with finitely supported perturbations.

math.SP↗

arXiv2009

An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schr{ö}dinger operator $ P(h)=-h^2Δ+V(x)$ in ${\mathbb R}^n$ such that the analytic potential $V$ has a non-degenerate critical point $x_0=0$ with critical value $E_0$ and we can define resonances in some fixed neighborhood of $E_0$ when $h>0$ is small enough. If the eigenvalues of the Hessian are $\zz$-independent the resonances in $h^δ$-neighborhood of $E_0$ ($δ>0$) can be calculated explicitly as the eigenvalues of the semi-classical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at $x_0.$ As a consequence, the resonances in a $h^δ$-neighborhood of $E_0$ determine the first $N$ terms in the Taylor series of $V$ at $x_0.$ The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.

math.AP↗

arXiv2007

Scattering Poles Near the Real Axis for Two Strictly Convex Obstacles

To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator $M$ along the trapped ray between the two obstacles. Using this method Ikawa and G{é}rard established the existence of parallel rows of poles in a strip $Im z\leq c$ as $Re z$ tends to infinity. Assuming that the boundaries are analytic and the eigenvalues of Poincar{é} map are non-resonant we use the Birkhoff normal form for $M$ to improve this result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.

math.AP↗

arXiv1995

Proof of the Strong Scott Conjecture for Atomic and Molecular Cores

The strong Scott conjecture about the electron density at a distance 1/Z from an atomic nucleus of charge $Z$ and its generalization for molecules are proved. The density, suitably scaled, converges to an explicit limiting density as $Z \to \infty$. Both a weak convergence and a pointwise convergence on spheres holds.

cond-mat↗