Peter Kuchment

Quantum graphs: I. Some basic structures

A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

Mathematics of thermoacoustic tomography

The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography.

Graph models for waves in thin structures

Abstract A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Band-gap structure of spectra of periodic dielectric and acoustic median I: scalar model

We investigate the band-gap structure of the spectrum of second-order partial differential operators associated with the propagation of waves in a periodic two-component medium. The medium is characterized by a real-valued position-dependent periodic function

Band-gap structure of spectra of periodic dielectric and acoustic media II: two-dimensional photonic crystals

\varepsilon ( x )

On the Spectra of Carbon Nano-Structures

that is the dielectric constant for electromagnetic waves and mass density for acoustic waves. The imbedded component consists of a periodic lattice of cubes where

Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed

\varepsilon ( x ) = 1

Spectral properties of classical waves in high-contrast periodic media

. The value of

On local tomography

\varepsilon ( x )

A Range Description for the Planar Circular Radon Transform

on the background is assumed to be greater than 1. We give the complete proof of existence of gaps in the spectra of the corresponding operators provided some simple conditions imposed on the parameters of the medium.