Invisibility and PT-symmetry

For a general complex scattering potential defined on a real line, we show that the equations governing invisibility of the potential are invariant under the combined action of parity and time-reversal (PT) transformation. We determine the PT-symmetric an well as non-PT-symmetric invisible configurations of an easily realizable exactly solvable model that consists of a two-layer planar slab consisting of optically active material. Our analysis shows that although PT-symmetry is neither necessary nor sufficient for the invisibility of a scattering potential, it plays an important role in the characterization of the invisible configurations. A byproduct of our investigation is the discovery of certain configurations of our model that are effectively reflectionless in a spectral range as wide as several hundred nanometers.Comment: 11 pages, 3 figures, revised version, accepted for publication in Phys.Rev.

Analyticity and uniform stability in the inverse spectral problem for Dirac operators

We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.Comment: 19 page

Detailed study of galaxies with the stellar counter-rotation phenomenon

The process of galactic disc growing is still not fully understood. In the majority of disk galaxies the gas and stars are located in the same plane and rotate in the same direction. However, there are kinematically peculiar galaxies hosting two counter-rotating stellar discs. Their origin is believed to be the result of a past event of accretion of gas followed by star formation. By studying such galaxies we can learn how much material, when, and how, have fallen onto the progenitor galaxy. We identified a sample of 56 counter-rotating galaxies in the MaNGA IFU survey and initiated a follow-up observing campaign at the 6-m telescope (BTA) aiming to determine the stellar population properties of both stellar discs. Our preliminary results suggest the dichotomy of the sample of counter-rotating galaxies. We found that most massive galaxies have extended counter-rotating disks, whose contribution to luminosity is higher than in the less massive galaxies suggestive of different evolutionary paths.Comment: 8 pages, 3 figure, 1 table; accepted for publication in the proceedings of the The Multifaceted Universe: Theory and Observations - 2022 (MUTO-2022) conference (published by PoS

On Necessary And Sufficient Conditions For Obtaining The Bases Of Banach Spaces

In the paper we cite necessary and sufficient conditions for completeness, minimality and basicity of systems in a Banach space that are obtained from initial system by changing the finite number of elements

Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a wave guide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic wave guide.Comment: Published versio

Asymptotics Of Eigenvalues And Eigenfunctions Of A Discontinuous Boundary Value Problem

The boundary problem is considered which occurs in the theory of small transversal vibrations of a smooth inhomogeneous string. The ends of the string assumed to be fixed and the midpoint of the string is damped by a pointwise force. The asymptotic behavior of eigenvalues and eigenfunctions of the considered boundary value problem is studied

EVALUATION OF THE APRICOT GENE POOL IN THE SOUTHERN URALS

Presented here are the results of studying the apricot gene pool at the South Ural Research Institute of Horticulture and Potato Cultivation (YUNIISK) in the context of the most important economically valuable traits. Varieties and forms of apricot have been identified for their high winter hardiness, fruit quality, and maximum adaptability to major biotic and abiotic factors of the environment -traits of great interest for further breeding work

The effect of abiotic factors on apricot yield in the Southern Urals

The productivity of apricot plantations in the Urals is limited by the irregularity of fruiting due to poor fruit bud winter hardiness in a majority of introduced cultivars. In 1999–2016, a research project was underway at the Department of Horticulture, Southern Ural Research Institute of Horticulture and Potato Growing, aimed at studying the apricot collection accessions of diverse geographic origin under the climate conditions of the Southern Urals. Such a long-term study made it possible to identify abiotic factors affecting the yield of this crop as well as to select highly adaptive and high-yielding cultivars for the Southern Urals: ‘Prizer’ (9.4 kg/tree), ‘Snezhinsky’ (9.2 kg/tree), ‘Kichiginsky’ (9.0 kg/tree) and ‘Uralets’ (8.5 kg/tree). Among the studied genotypes there were tree forms with irregular fruiting pattern, characterized by low resistance to adverse effects of abiotic factors: ‘Khabarovsky’ (0.9 kg/tree) and ‘Michurinsky No. 22’ (0.3 kg/tree). In the environments of the Southern Urals, critical winters (continuous frosts of –40°C) considerably damage generative buds, which results in having no apricot harvest in such years. Despite the harsh winters, annual shoots of local apricot-trees tended to freeze only to a small degree. Field survey of the apricot plantations showed that generative buds of local apricot cultivars (‘Kichiginsky’, ‘Prizer’, ‘Snezhinsky’ and ‘Uralets’) could withstand frosts of –40... –43°С (2003) only if they were brief, while continuous frosts destroyed them completely (2006, 2010). In addition, a decline in harvest was observed as a result of springtime frosts and temperature fluctuations in the end of winter (1999, 2014). Despite the abundant flowering in 2001, 2015 and 2016, the yield of apricot trees was low due to the frosts during the flowering period. Productivity of apricot trees also depends on their genetic characteristics, and in particular, on the geographical origin of cultivars. The introduced cultivars ‘Khabarovsky’ and ‘Michurinsky No. 22’ yielded fruit only thrice during the entire period of research

Inverse spectral problems for Dirac operators with summable matrix-valued potentials

We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.Comment: 32 page

The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result