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This paper deals with the Sturm-Liouville problem that feature distribution potential, polynomial dependence on the spectral parameter in the first boundary condition, and analytical dependence, in the second one. We study an inverse spectral problem that consists in the recovery of the potential and the polynomials from some part of the spectrum. We for the first time prove local solvability and stability for this type of inverse problems. Furthermore, the necessary and sufficient conditions on the given subspectrum for the uniqueness of solution are found, and a reconstruction procedure is developed. Our main results can be applied to a variety of partial inverse problems. This is illustrated by an example of the Hochstadt-Lieberman-type problem with polynomial dependence on the spectral parameter in the both boundary conditions.

The static and dynamic properties of spin-polarons in La-doped $CaMnO_3$ are explored theoretically, by means of an effective low energy Hamiltonian. All parameters from the Hamiltoniain are evaluated from first principles theory, without adjustable parameters. %We compare different geometries, like spin-polarons in bulk, surface and as single two-dimensional layers. The Hamiltonian is used to investigate the temperature stability as well as the response to an external applied electric field, for spin-polarons in bulk, surface and as single two-dimensional layers. Technically this involves atomistic spin-dynamics simulations in combination with kinetic Monte Carlo simulations. Where a comparison can be made, our simulations exhibit an excellent agreement with available experimental data and previous theory. Remarkably, we find that excellent control of the mobility of spin-polarons in this material can be achieved, and that the critical parameters deciding this is the temperature and strength of the applied electrical field. We outline different technological implications of spin-polarons, and point to spin-polaronics as an emerging sub-field of nano-technology. In particular, we demonstrate that it is feasible to write and erase information on atomic scale, by use of spin-polarons in $CaMnO_3$., Comment: 13 pages, 8 figures

We consider second order linear differential operators possessing a term depending on the unknown function with a fixed argument and study the uniqueness of recovering the operators from the spectrum. We also obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability., Comment: 14 pages

Multipolaron solutions were studied in the framework of the Holstein one-dimensional molecular crystal model. The study was performed in the continuous limit where the crystal model maps into the nonlinear Schr\"odinger equation for which a new periodic dnoidal solution was found for the multipolaron system. In addition, the stability of the multi-polaron solutions was examined, and it was found that cnoidal and dnoidal solutions stabilize in different ranges of the parameter space. Moreover, the model was studied under the influence of nonlocal effects and the polaronic dynamics was described in terms of internal solitonic modes., Comment: 5 pages, 2 figures and Supplementary information

$La_xCa_{1-x}MnO_3$ (LCMO) has been studied in the framework of density functional theory (DFT) using Hubbard-U correction. We show that the formation of spin-polarons of different configurations is possible in the G-type antiferromagentic phase. We also show that the spin-polaron (SP) solutions are stabilized due to an interplay of magnetic and lattice effects at lower La concentrations and mostly due to the lattice contribution at larger concentrations. Our results indicate that the development of SPs is unfavorable in the C- and A-type antiferromagnetic phases. The theoretically obtained magnetic state diagram is in good agreement with previously reported experimental results, Comment: 6 pages, 4 figures

Electron localization and polaron mobility in oxygen deficient as well as Li doped monoclinic tungsten trioxide have been studied. We show that small polarons formed in the presence of oxygen vacancy prefer the bipolaronic $W^{5+}- W^{5+}$ configuration whereas the $W^{6+}- W^{4+}$ configuration is found to be metastable. Our calculations suggest that bipolarons are tightly bound by the vacancy and therefore largely immobile. On the contrary, polarons formed as a result of Li intercalation can be mobile, the activation energy for polaron jumping in this case varies between 98 and 124 meV depending on the crystallographic direction. The formation of $W^{5+}- W^{5+}$ bipolarons in $Li-WO_{3}$ is possible. When situated along $[001]$ the bipolaronic configuration is 8 meV lower in energy than two separate $W^{5+}$ polarons., Comment: 7 pages, 6 figures