# Research topics

The laboratory's activities are devoted to the development of new mathematical and computer methods for studying natural science models that are relevant and promising for the development of high technologies that are actively studied in the world science.

### Research directions

- Study of parabolic problems (including for pseudodifferential equations), construction of various representations for fundamental solutions and solutions to the Cauchy problem.
- Development of methods for justifying asymptotic solutions in spaces with exponential weight (problems of large compactions in the theory of random processes).
- Methods for solving problems of quantum theory and the theory of random processes on grids by reducing them to pseudodifferential equations, constructing and justifying asymptotic solutions.
- Development of new methods for numerical solution of Sobolev-type problems arising when modeling processes in microelectronics.
- Theoretical modeling of quantum and collective phenomena in low-dimensional materials and nanostructures.

Theoretical studies and computer simulation of quantum, electronic, and optical phenomena are carried out in micro- and nanostructures, new two-dimensional materials and heterostructures composed of them.

Main results in this area:

● The extremely high sensitivity of the Bose-Einstein condensate of excitonic polaritons in an optical resonator made of an organic polymer has been demonstrated, reaching the fundamental limit: the entry of even a single photon into the resonator leads to a noticeable enhancement of the luminescent radiation coming from it.

● A new mathematical description of the quantum state of a many-particle system of excitonic polaritons - hybrid excitations of light and matter - is proposed, based on recurrent relations derived from the condition of approximate conservation of the number of polaritons. This description makes it possible to simulate the dynamics of a polariton system, the establishment of thermal equilibrium and Bose-Einstein condensation in it, and calculate the spectrum of luminescent radiation and its optical coherence.

● The Andreev-Bashkin effect, or superfluid drag effect, between a superfluid Bose-Einstein condensate of excitonic polaritons in a microcavity and a superconducting two-dimensional electron gas is predicted.

● The Hartree-Fock-Bogolyubov theory was constructed for the Bose-Einstein condensation of excitonic polaritons at a finite temperature, which explains the experimentally observed deviations of the quasiparticle velocity from the predictions of the standard Bogolyubov theory.

● Unusual quantum states of pseudomagnetoexcitons are predicted - bound pairs of electrons and holes that appear in a graphene layer subjected to mechanical stretching, in which an analogue of a magnetic field acts on electrons and holes.

● The possibility of establishing quantum entanglement between two or three qubits interacting with the same electromagnetic mode has been demonstrated, due to photon leakage - a purely dissipative process.

● The non-Markovian quantum dynamics of systems of spins, qubits and excitonic polaritons interacting with a thermal reservoir, which has its own delayed quantum dynamics, has been studied. It is shown that non-Markov effects can radically change the dynamics of the Bose-Einstein condensate of excitons: for example, the establishment of nonlinear relaxation oscillations or completely chaotic dynamics is possible.

● A generalized virial theorem for massless Dirac electrons (such as electrons in graphene or topological Weyl semimetals) was obtained, relating the average kinetic energy of electrons, the energy of their Coulomb interaction and the pressure of the electron gas at the boundary. The connection between the virial theorem and the behavior of the wave function near the material boundary is demonstrated.

- Methods of semiclassical approximation for difference and differential equations.

Main results in this area:

● An approach to the use of semiclassical approximation methods for solving difference and differential equations is being developed.

● Uniform semiclassical asymptotics of solutions to a second-order difference equation for large values of the argument are constructed. The proved theorem was applied to construct uniform asymptotics of Laguerre polynomials for large values of the argument and the degree of the polynomial, which generalized the scope of application of the known asymptotics of Laguerre polynomials constructed on an interval.

● Oscillating semiclassical asymptotics of the tunnel splitting of the upper and lower energy levels of the spectrum of quadratic operators on the Lie algebra su(1,1) are constructed.

- Structural network analysis of available data on the connectome and human transcriptome, including the development of new analysis methods based on spectral characteristics; phase transitions and critical phenomena in exponential and geometric graphs, and application of models to the data description.

Main results in this area:

● Numerical and analytical description of Anderson localization in partially unordered randomly regular graphs.

● Identifying k-clique percolations in human structural connectomes and constructing models that reproduce observed features.

● Development of new methods for identifying clusters in human gene-regulatory networks.

- Algebraic methods for studying quantum models with non-commutative symmetry algebras.

The research is aimed at studying non-Lie algebras with a finite number of generators, describing their irreducible representations and constructing the spectral theory of these algebras.

From the algebraic point of view, there are two main questions: first, how to construct coherent states and the corresponding irreducible representations of such algebras; second, how to relate these quantum representations to some classical symplectic leaves in a Poisson manifold. An additional task is to establish a connection between irreducible representations, coherent states and special functions that arise during their construction.

From the physical point of view, the main problem of the research is to study algebras that naturally arise (as symmetry algebras) in various quantum mechanical models. The irreducible representations and coherent states of these algebras play a decisive role in the spectral analysis of quantum problems. Here an algebraic approach is developed, consisting of the successive application of the methods of operator averaging and coherent transformation. This approach becomes the key point for studying quantum models with strong degeneracy of the spectrum of the leading part of the operator (for example, due to resonance), since the standard perturbation theory does not work here.

Recently, in the framework of this direction of research, a new approach to solving spectral problems with continuous spectrum has been outlined. For generalized eigenfunctions of such problems, a new integral representation in terms of coherent distributions is proposed. Such distributions have all the key properties of coherent states but do not have a finite norm in Hilbert space. It is shown that they “work” better in the problems involving the continuous spectrum of operators than ordinary coherent states.

- Hydrodynamic multiscale flow problems with a double-deck boundary layer structure.

Using a combined numerical-asymptotic approach, problems of flow around small irregularities at large Reynolds numbers in various geometries are studied: - along a plate, inside a pipe, flows induced by a rotating disk. The developed approach allows one to avoid resource-consuming direct numerical modeling of the Navier-Stokes equations in a region with several spatial scales.

One of the latest results is the modeling of a three-dimensional problem of flow around a small localized irregularity.

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