### Article

## Асимптотические решения одномерного уравнения Хартри с негладким потенциалом взаимодействия. Асимптотика квантовых средних

We consider an eigenvalue problem for the two-dimensional Hartree operator with a small parameter at the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct an asymptotic expansion around the circle where the solution is localized.

Within the presented monograph for the first time statistical approaches, based on the self-consistent field theory, were presented for the theoretical description of the thermodynamic properties of the ion-molecular systems (electrolyte solutions, ionic liquids, dielectric polymers and metal-organic frameworks) in the bulk solution and at the interfaces with the account for their molecular structure. In the book one can also find a thorough analysis of the state of the art of the theory and modeling of the ion-molecular systems. The book can be used as a guideline for the physical-chemists, physicists and nanotechnologists, working in the area of theory and simulation of the ion-molecular systems, and can let them utilize the discussed approaches for the solving of the different tasks of the chemical thermodynamics and condensed matter physics. Therefore, the book is addressed to the specialists, working in the area of physical chemistry and condensed matter physics, as well as to senior students and PhD students of profile specialty.

We consider the eigenvalue problem for a perturbed two-dimensional

resonance oscillator. The excitation potential is given by a Hartree-type

nonlinearity with a smooth self-action potential. We use asymptotic

formulas for the quantum averages to obtain asymptotic eigenvalues and

asymptotic eigenfunctions near the lower boundaries of spectral clusters

which are formed near the energy levels of the unperturbed operator.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.