Abstract. We study integro-differential equations with unbounded operator coefficients on the half-line. The symbols of these equations are polynomial operator pencils perturbed by operator functions holomorphic (regular) in the left half-plane.Results concerning the Fredholm property of initial-boundary value problems for these equations on the half-line and the properties of their exponential (elementary) solutions are established.The properties of the derived chains constructed from the root vectors of the operator functions that are the symbols of these equations are considered.

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This review summarizes results of our study of the application of ion-beam assisted deposition (IBAD) technology for creation of nanoporous thin-film structures that can absorb more than 6 wt.% of hydrogen. Data of mathematical modeling are presented highlighting the structure formation and component creation of the films during their deposition at the time of simultaneous bombardment by mixed beam of nitrogen and helium ions with energy of 30 keV. Results of high-resolution transmission electron microscopy revealed that VNxfilms consist of 150–200 nm particles, boundaries of which contain nanopores of 10–15 nm diameters. Particles themselves consist of randomly oriented 10–20 nm nanograins. Grain boundaries also contain nanopores (3–8 nm). Examination of the absorption characteristics of VNx, TiNx, and(V,Ti)Nxfilms showed that the amount of absorbed hydrogen depends very little on the chemical composition of films, but it is determined by the structure pore. The amount of absorbed hydrogen at 0.3 MPa and 20°C is 6-7 wt.%, whereas the bulk of hydrogen is accumulated in the grain boundaries and pores. Films begin to release hydrogen even at 50°C, and it is desorbed completely at the temperature range of 50–250°C. It was found that the electrical resistance of films during the hydrogen desorption increases 104times.

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