Vol 224, No 1 (2017)

Interest in inverse dynamical, spectral, and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum applications. Recently a new effective leaf peeling method has been proposed by S. Avdonin and P. Kurasov for solving inverse problems on trees (graphs without cycles). It allows recalculating efficiently the inverse data from the original tree to smaller trees, “removing” leaves step by step up to the rooted edge. In this paper, the main step of the spectral and dynamical versions of the peeling algorithm, i.e., recalculating the inverse data for a “peeled tree” is described. Bibliography: 12 titles.

In a series of articles, V. S. Buldyrev considered interference head waves, called now Buldyrev waves. The main goal in the present paper is to obtain formulas describing Buldyrev waves using the locality principle. Similar formulas have been deduced (from other but also heuristic considerations) by Buldyrev earlier. A different point of view on the Buldyrev wave is described by V. M. Babich. Some formulas for waves of this class contain an illusory contradiction with the concept of locality. It is demonstrated that the contradiction is fictitious and the formulas deduced by Buldyrev and the results obtained later perfectly agree with the locality principle.

Simple explicit solutions of the homogeneous wave equation with constant propagation speed, having a power-like singularity at a moving space point, are constructed. The constructions are based on a complexified Bateman-type solution. An example of such a solution showing exponential decay with distance from the singular point is presented.

The present paper aims at announcing a new approach to the construction of the asymptotics (at infinity in configuration space) of the resolvent kernel of the Schrödinger operator in the scattering problem of three one-dimensional quantum particles interacting by compactly supported pair repulsive potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator can be constructed explicitly. It should be emphasized that the restriction of the consideration to the case of compactly supported pair potentials does not lead to a simplification of the problem in its essence, since the potential of the interaction of all three particles remains nondecreasing at infinity, but allows one to put aside a certain number of technical details.

The paper is devoted to ray type solutions for waves of finite deformation in nonlinear, physically linear elastic media. These waves are generalizations of Bland plane waves for isotropic nonlinear media. For the waves considered, the fast oscillation and slow oscillation parts interact in the process of propagation. The forms of waves change adiabatically. An example of plane waves in inhomogeneous elastic media is considered.

Transmission conditions at a bifurcation point in a one-dimensional model of blood vessels are derived by using a three-dimensional model. Both classical Kirchhoff conditions ensuring the continuity of pressure and the vanishing of the flux should be modified in order to reflect properly the elastic properties of blood vessels. A simple approximate method of calculation of new physical parameters in the transmission conditions is proposed. Simplified models of straight sections of arteries with localized defects such as micro-aneurysms and cholesterol plaques are developed; these models also require the statement of some transmission conditions. Bibliography: 34 titles.

Some generalizations of the Fourier and Hartley transforms on ℝ and ℝⁿ are studied. These transforms involve functional coefficients. The Parseval identities, convolution formulas, conditions of self-adjointness and unitary property for new transforms are obtained.

The difference Schrӧdinger equation ψ(z+h)+ψ(z−h)+v(z)ψ(z) = Eψ(z), z ∈ ℂ, is considered, where h > 0 and E ∈ ℂ are parameters and v is a function analytic in a bounded domain D ⊂ ℂ. An asymptotic method is developed for studying its solutions in the domain D for small positive h. Bibliography: 4 titles.