Quantum-mechanical model of dielectric losses in nanometer layers of solid dielectrics with hydrogen bonds at ultra-low temperatures

Upon based the finite difference methods construct the solutions for Liouville quantum kinetic equation linearized by the external field, in complex with the stationary Schrodinger equation and the Poisson operator equation, for an ensemble of non-interacting hydrogen ions (protons) migrating in the field of a crystal lattice perturbed by a variable polarizing field. The influence of the phonon subsystem is not taken into account. The equilibrium (non-balanced) proton density matrix is calculated using quantum Boltzmann statistics. The temperature spectra of dielectric losses tangent angle for hydrogen bonded crystals (HBC) in a wide temperature range (50–550 K) are calculated. At the theoretical level detected the effects of nano-crystalline states (1–10 nm) during the polarization of HBC in the region of ultra-low temperatures (4–25 K).

Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds

It is proved that the Liouville function associated with the semilinear equation $\Delta u -g(x,u)=0$ is identical to zero if and only if there is only a trivial bounded solution of the semilinear equation on non-compact Riemannian manifolds. This result generalizes the corresponding result of S.A. Korolkov for the case of the stationary Schrödinger equation $ \Delta u-q (x) u = 0$. The concept of the capacity of a compact set associated with the stationary Schrödinger equation is also introduced and it is proved that if the capacity of any compact set is equal to zero, then the Liouville function is identically zero.

Quasistationary states in a quantum dot formed at the edge of a topological insulator by magnetic barriers with finite transparency

A model of quasistationary states is constructed for the one-dimensional edge states propagating along the edge of a two-dimensional topological insulator based on HgTe/CdTe quantum well in the presence of magnetic barriers with finite transparency. The lifetimes of these quasistationary states are found analytically and numerically via different approaches including the solution of the stationary Schrödinger equation with complex energy and the solution of the transmission problem for a double barrier structure. The results can serve as a guide for determining the parameters of magnetic barriers creating the quantum dots where the lifetimes for the broadened discrete levels are long enough for manipulation with their occupation numbers by external fields.