Spectral bounds for the Hellmann potential

The method of potential envelopes is used to analyse the bound state spectrum of the Schroedinger Hamiltonian H=-\Delta+V(r), where the Hellmann potential is given by V(r) = -A/r + Be^{-Cr}/r, A and C are positive, and B can be positive or negative. We established simple formulas yielding upper and lower bounds for all the energy eigenvalues.Comment: 9 pages, 2 figures, typos correcte

Micromorphological features of the leaf epidermis of the evening primrose cultivars of the VILAR biocollection

Received: May 8th, 2021 ; Accepted: June 19th, 2021 ; Published: June 22th, 2021 ; Correspondence: [email protected] of the work. For the first time, a comparative study of the epidermis of the leaf plates of four cultivars of Oenothera biennis L. was carried out. The aim of the work. Study and comparison of morpho-anatomical signs of the epidermis of the leaf plate of the cultivars of evening primrose to reveal their potential ecological plasticity. Materials and methods. There were used leaves of plants of the second year of life in the phase of mass flowering. Micromorphological study of the epidermis of leaf plates included: determination of the shape of the main cells of the epidermis, the size and number of stomata per 1 mm2 , the type of stomatal apparatus, the presence and parameters of trichomes. The number of trichomes was calculated per 1 mm2 . Results. In the Genoteros cultivar, the stomata were smaller; their number on the lower epidermis was the lowest. Two types of trichomes were classified. At cultivar Genoteros unicellular covering trichomes were mainly located in the upper epidermis of the leaf. No covering trichomes were found in the Svetlyachok cultivar. The Tverskoj cultivar differs in the length of the hairs: from 774 to 790 microns. Papillary single-celled trichomes on the leaves of the studied cultivar of evening primrose are concentrated on both sides of the epidermis, mostly in the middle part of the leaf, less often - in the mesophyll space. Conclusion. Based on the obtained data, the authors established the potential ecological plasticity of the studied varieties of evening primrose for the subsequent study of the influence of stress factors on the described cultivars. The results obtained may be important taxonomic for the correct identification of problematic Onagraceae taxa and in further breeding work