Alexei Iantchenko

Resonances for periodic Jacobi operators with finitely supported perturbations

We describe the resonances and the eigenvalues of a periodic Jacobi operator with finitely supported perturbations. In the case of small diagonal perturbations we determine their asymptotics.

Periodic Jacobi operator with finitely supported perturbation on the half-lattice

Maths2010 British Mathematical Colloquium and British Applied Mathematics Colloquium Apr 6, 2010- Apr 9, 2010, Edinburgh, UK

Resonance spectrum for one-dimensional layered media

We consider the “weighted” operator Pk= − ∂x a(x)∂ x on the real line with a step-like coefficient which appears when propagation of waves through 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 Pk. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of Pk has band structure. In the article, we study a transition to semi-infinite medium by taking the limit k→ ∞ . The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k − 1 or k resonances in each band. We prove that as k→ ∞ , the resonance spectrum converges to the real axis.

Scattering Poles near the Real Axis for Two Strictly Convex Obstacles

Abstract.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 Gérard (cf. [7]) obtained complete asymptotic expansions for the poles in a strip Im z ≤ c as Re z tends to infinity. He established the existence of parallel rows of poles close to

Resonance Spectrum for a Continuously Stratified Layer with Application to Ultrasound Testing of Articular Cartilage

\frac{\pi k}{d}+ ij\delta,\, k \in {\mathbb{Z}},\, j \in {{\mathbb{Z}}_{+}}.

Quasi-normal modes for Dirac fields in the Kerr-Newman-de Sitter black holes

Assuming that the boundaries are analytic and the eigenvalues of Poincaré map are non-resonant we use the Birkhoff normal form for M to improve his result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.

Resonance spectrum for a continuously stratified layer: application to ultrasonic testing

A large amplitude oscillatory shear (LAOS) is considered in the strain-controlled regime, and the interrelation between the Fourier transform and the stress decomposition approaches is established. Several definitions of the generalized storage and loss moduli are examined in a unified conceptual scheme based on the Lissajous–Bowditch plots. An illustrative example of evaluating the generalized moduli from a LAOS flow is given.

Resonances for 1D massless Dirac operators

Ultrasound wave propagation in the articular cartilage layer is considered. The cartilage is modeled by a nonhomogeneous linearly elastic layer of constant thickness. The resonances for the corresponding acoustic propagator are studied. It is shown that the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. The spacing between two successive resonances turns out to be sensitive to articular cartilage degeneration.