The object of this project is to study the mathematical models of nanosystems of different types.

The nanosystem states characterized by critical specific characteristics such as the resonance of the frequencies of the Hamiltonian higher-order part, nontrivial curvature and sharpened points, tunneling and multilocalization of states in multiply connected domains of the phase space are considered. Until now, the properties of such nanosystems have been studied rather poorly, and this can be explained by the following: the mathematical structures that can be used to describe their states were not known until recently, although these problems were originally posed at the time of the quantum mechanics origination, while similar open problems, for example, in electrodynamics and acoustics, were also posed long ago. One of the most significant difficulties is caused by a high or even infinite multiplicity of the spectrum degeneration for the higher-order or “adiabatically stable” part of the Hamiltonian nanosystem. This makes the usual matrix methods of the perturbation theory of Rayleigh-Schroedinger type inefficient and does not permit an understanding of the relationship through classical mechanics and using the powerful geometric apparatus in the semiclassical limit. The goal of the project is to overcome these difficulties.

The empirical basis of the studies in this project is formed by physical nanomodels described in the relevant scientific literature.

The phenomena of tunneling and multilocation of states in double potential wells and the Gibbs paradox related to these phenomena are used in the project. The mode tunneling is described in some model examples and in the case of a double nonsymmetric potential well of general form. Some conditions under which the bilocation of states arises in a double nonsymmetric well are obtained.

New general phenomena of resonance assembly and resonance intertwining in nanosystems are described. The existence conditions for the state geometry assembly are obtained. The resonance assembly of states for a hydrogen-like impurity center on a surface in twisted magnetic and electric fields is described.

The two-frequency resonances in Penning quantum nanotraps are described. The operator averaging in two orders of the perturbation introduced by the linear inhomogeneity of the magnetic field (the Ioffe field) is performed. Generators of the algebra of symmetries for the resonance 3:(-1) in the directions orthogonal to the trap axis are constructed. The expression for the effective Hamiltonian in terms of generators of the algebra of symmetries is obtained.

The results of this project can be used to design nanodevices of different types.

The results of the studies will be used in the educational process at the Chair of Applied Mathematics MIEM HSE in the profile “Nanosimulation’’. The methods, algorithms, and mathematical models developed in the project will underpin the course tasks and diploma projects of students at the Chair of Applied Mathematics MIEM HSE and will be used in lecture courses delivered at the chair. The results already obtained in these investigations and further studies based on them will be used in Candidate of Science dissertations prepared by the post-graduate students.