Inverse Scattering Problem on the Axis for the Schrödinger Operator with Triangular 2 x 2 Matrix Potential. I. Main Theorem

The necessary and sufficient conditions for solvability of ISP under consideration are obtained

Inverse Scattering Problem on the Axis for the Schrödinger Operator with Triangular 2 x 2 Matrix Potential. II. Addition of the Discrete Spectrum

The theorem of the necessary and sufficient conditions for the solvability of ISP under consideration is proved. The method of addition of the discrete spectrum to the considered matrix not self-adjoint case is developed

Necessary and Sufficient Conditions in Inverse Scattering Problem on the Axis for the Triangular 2 x 2 Matrix Potential

The characteristic properties of the scattering data for the Schr¨odinger operator on the axis with a triangular 2 × 2 matrix potential are obtained. A difference between the necessary and sufficient conditions for solvability of ISPunder consideration, contained in the previous works of the authors, is eliminated

On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients

For the Sturm-Louville equation with block-triangular matrix potential that increases at infinity, both increasing and decreasing at infinity matrix solutions are found. The structure of spectrum for the differential operator with these coefficients is defined.Для уравнения Штурма-Лиувилля с блочно-треугольным растущим на бесконечности матричным потенциалом построены убывающие и растущие на бесконечности матричные решения. Установлена структура спектра дифференциального оператора с такими коэффициентами

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

The velocity of slow nuclear burning in the two-group approximation

The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are presented

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Escherichia coli RuvBL268S: A mutant RuvB protein that exhibits wild-type activities in vitro but confers a UV-sensitive ruv phenotype in vivo

The RuvABC proteins of Escherichia coli process recombination intermediates during genetic recombination and DNA repair. RuvA and RuvB promote branch migration of Holliday junctions, a process that extends heteroduplex DNA. Together with RuvC, they form a RuvABC complex capable of Holliday junction resolution. Branch migration by RuvAB is mediated by RuvB, a hexameric ring protein that acts as an ATP-driven molecular pump. To gain insight into the mechanism of branch migration, random mutations were introduced into the ruvB gene by PCR and a collection of mutant alleles were obtained. Mutation of leucine 268 to serine resulted in a severe UV-sensitive phenotype, characteristic of a ruv defect. Here, we report a biochemical analysis of the mutant protein RuvBL268S. Unexpectedly, the purified protein is fully active in vitro with regard to its ATPase, DNA binding and DNA unwinding activities. It also promotes efficient branch migration in combination with RuvA, and forms functional RuvABC-Holliday junction resolvase complexes. These results indicate that RuvB may perform some additional, and as yet undefined, function that is necessary for cell survival after UV-irradiatio