The solutions of Heisenberg equations and two-particles eigenvalue problems for nonrelativistic models of current-current fermion interaction and N, Θ model are obtained in the frameworks of dynamical mapping method. The equivalence of different types of dynamical mapping is shown. The connection between renormalization procedure and theory of selfadjoint extensions is elucidated. General considerationThe main problem of QFT follows from the fact that any solutions of Heisenberg equations (HE) are the operator distributions which products, always appearing in that equations, are ill-defined.So, the correct definition of field equations (and Hamiltonian itself) implies some knowledge about qualitative properties of its solutions which in their turn depend on the form of these equations by a very singular manner. The usual way to go out from this closed circle is connected with perturbation theory. It is based on the assumption that product of Heisenberg fields (HF) may be defined as well as for the free ones and solution of HE may be obtained by perturbation in the Fock space of renormalized free fields. However, it is impossible on such a way to work with nonrenormalizable theory and to understand the origin of the bound states. We consider another possibility which is based on the idea of dynamical mapping and reduce the product of HF to the normal ordering for the product of the physical fields. It is originated from the works of R.Haag In this approach the problem of making a sense for formal expression of HF:for Hamiltonian given as a functional H = H [Ψ( x, t)], is divided on two parts. The first one is the construction of the following operator realization of the initial fields Ψ( x, t 0 ) = Ψ[ψ] via physical fields ψ( x, t) ≡ ψ A α ( k) , which, on the one hand, should be consistent with CCR (CAR) (α = 1, 2) {Ψ α ( x, t) , Ψ β ( y, t)} = 0 = {ψ α ( x, t) , ψ β ( y, t)}, {Ψ α ( x, t) , Ψ † β ( y, t)} = δ 3 ( x − y) δ αβ = {ψ α ( x, t) , ψ † β ( y, t)}, {A α ( k) , A β ( q)} = 0; {A α ( k) , A † β ( q)} = δ 3 ( k − q) δ αβ ,

A: The TAIGA observatory addresses ground-based gamma-ray astronomy at energies from a few TeV to several PeV, cosmic ray physics from 100 TeV to several EeV as well as for search for axion-like particles, Lorentz violations and another evidence of New Physics. In 2020 year a one square kilometer TAIGA setup should be put in operation.

The Tunka-Grande and TAIGA-Muon arrays are the part of a single experimental complex, which also includes the Tunka-133 and TAIGA-HiSCORE (High Sensitivity COsmic Rays and gamma Explorer) wide-angle Cherenkov arrays, TAIGA-IACT array (Imaging Atmospheric Cherenkov Telescope) and Tunka-Rex radio antennas array (Tunka Radio Extension). This complex is located in the Tunka Valley (Buryatia Republic, Russia), 50 km from Lake Baikal. It is aimed at investigating the energy spectrum and mass composition of charged cosmic rays in the energy range 100 TeV - 1000 PeV, searching for diffuse gamma rays above 100 TeV and studying local sources of gamma rays with energies above 30 TeV. This report outlines 3 key points. The first is a description of the Tunka-Grande and TAIGA-Muon scintillation arrays. The second part presents preliminary results of the search for diffuse gamma rays with energies above 50 PeV according to the Tunka-Grande data. The third part is devoted to the prospects of the search for diffuse gamma rays with energies above 100 TeV using the TAIGA-Muon array.

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