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

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

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

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

Multidimensional Borg-Levinson Theorem

We consider the inverse problem of the reconstruction of a Schr\"odinger operator on a unknown Riemannian manifold or a domain of Euclidean space. The data used is a part of the boundary $\Gamma$ and the eigenvalues corresponding to a set of impedances in the Robin boundary condition which vary on $\Gamma$. The proof is based on the analysis of the behaviour of the eigenfunctions on the boundary as well as in perturbation theory of eigenvalues. This reduces the problem to an inverse boundary spectral problem solved by the boundary control method

ENVIRONMENTAL PLASTICITY OF VARIOUS PLUM CULTIVARS UNDER THE CONDITIONS OF CHELYABINSK PROVINCE

The use of adaptable fruit and berry cultivars significantly increases the environmental sustainability of horticulture. In 2014–2018, the assortment of plum in the Urals was evaluated using the parameters of productivity, environmental plasticity and stability under the conditions of Chelyabinsk Province. The analysis of environmental plasticity and stability helped to identify adaptable cultivars of Chinese plum (Prúnus salicina L.): ‘Altayskaya yubileynaya’ and ‘Uralskaya zolotistaya’. Plastic plum cultivars included cv. ‘Uralskaya zolotistaya’ (yield: 5.62 t/ha; bi = 1.10; Si2 = 25.7), ‘Uralskaya serebristaya’ (5.53 t/ha; 1.16; 21.3) and ‘Manchzhurskaya krasavitsa’ (5.53 t/ha; 1.21; 33.9); their productivity varied in accordance with changes in environmental conditions. Intensive-type cultivars with high responsiveness to the improvement of growing conditions (bi significantly higher than 1) were cvs. ‘Uvelskaya’ (5.62 t/ha; 1.46; 26.8) and ‘Krasnoselskaya’ (5.04 t/ha; 1.35; 45.7). Cv. ‘Zhemchuzhina Urala’ (4.65 t/ha; 0.05; 22.8) belongs to the cultivars with low plasticity (the bi value close to zero); it is characterized by a weak response to a change in environmental conditions. Cv. ‘Altayskaya yubileynaya’ produces the highest yield (6.16 t/ha) due to its plasticity (bi = 0.91), but has low stability (Si2 = 102.5), while cv. ‘Shershnevskaya’ secures rather high productivity (5.23 tons per hectare) due to high stability (Si2 = 32.7) and medium responsiveness to changes in environmental conditions (bi = 0.75)

Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.Comment: 39 pages, uses iopart.cls, iopams.sty and setstack.sty by IO

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

Inverse spectral problems for energy-dependent Sturm-Liouville equations

We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.Comment: AMS-LaTeX, 28 page