Inverse problems for matrix Sturm-Liouville operators

Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Dedicated to the memory of B. M. Levitan     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

On recovering Sturm—Liouville operators on graphs

Sturm—Liouville differential operators on compact graphs are studied. We establish properties of the spectral characteristics and investigate three inverse problems of recovering the operator from the so-called Weyl functions, from discrete spectral data, and from a system of spectra. For these inverse problems, we prove uniqueness theorems and obtain procedures for constructing the solutions by the method of spectral mappings.     

Half-Range Expansions for an Astrophysical Problem

Problems of spectral analysis are studied for an indefinite singular boundary value problem coming from astrophysical theory of particle acceleration around shocks. This leads to a nonclassical initial-boundary value problem for a partial differential equation that can bereduced by separation of variables to an indefinite Sturm–Liouville problem for which we establish Riesz basis properties of the eigen- and associated functions and formulate completeness and expansion theorems.     

Synthesis and characterization of poly[phenyl(fluoroalkyloxaalkane)silanediyl]s

Monomeric fluoroalkoxyalkyl(phenyl)dichlorosilanes were synthesized with quantitative yields by hydrosilylation of allyl ethers of fluoroalcohols with phenyldichlorosilane with Pt(Ph₃Sb)₂Cl₂ complex as a catalyst. Starting from these monomers, corresponding polysilanes with weight‐average molecular weights of 2–3 × 10⁴ were obtained by the Wurtz reaction. Increasing the CF₂ groups containing two to six monomer units caused the char yields to increase from 3 to 48% at 800 °C in air. The emission characteristics of these polysilanes in a chloroform solution were examined. They had a narrow peak at 344 nm with a small Stokes shift (≈20 nm). © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 3761–3767, 2003     

Recovering nonselfadjoint differential pencils with nonseparated boundary conditions

Nonselfadjoint second-order differential pencils on a finite interval with nonseparated boundary conditions are studied. We establish some important properties of spectral characteristics and investigate inverse problems of recovering the operator from its spectral data. For these inverse problems, we prove the corresponding uniqueness theorems and provide procedures for constructing their solutions.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

Calculation of the composition of the liquid and solid phases in the Ga-Al-As system cooled with evaporation of the volatile component

The regulation of the composition of epitaxial films of solid solutions of compounds obtained in liquid-phase epitaxy is discussed. A new method is proposed for controlling the composition of an epitaxial film based on crystallization by cooling of the molten solution on a substrate with simultaneous change in the composition of the liquid phase due to evaporation of a volatile component. A calculation has been made and the possibilities of such a process demonstrated for the example of Ga_yAl_1–yAs solid solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 80–83, September, 1980.