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The state policy of health preservation of Russians and the process of introducing a healthy lifestyle into their everyday life is hampered by the lack of sufficient self-activity and purposefulness of the individual ecological and valeological behavior of representatives of various population groups. According to the authors of the article, one of the important indicators of the maturity of professional and labor competencies of school and student youth is their readiness and desire for permanent self-preserving behavior. “With numbers in hand,” the authors show the scale of deviant deviations and the phenomena of spontaneous irresponsibility in the educational and leisure activities of students, hindering the preservation and development of physical culture, the accumulation and effective use of their psychophysiological and labor potential. The conclusions of the proposal of the authors of the article are based on the results of a number of sociological surveys conducted in 2000-2020. at the Institute of Economics of the Ural Branch of the Russian Academy of Sciences in a number of secondary schools and universities of the Ural and Volga Federal Districts.

In the article, relevant problems of social and professional health of teachers of the Ural higher education institutions are analyzed. The author recognizes that the higher education is created as an interaction between participants of educational process, each of which at the same time acts both as the subject pursuing the valid aims, and as object of orientation for other individuals of the environment of the immediate environment. On the basis of statistical data, the author shows that in socio-economic behavior of the vast majority of students of the Ural higher education institutions are inherent as common features of deviant subculture of youth (addiction to alcohol and drugs, violence manifestations, prostitution etc.), and special, connected with educational process (admissions of occupations, use of cribs, plagiarism when performing term papers, diplomas, roughness and tactlessness in relation to teachers, etc.). According to the author, the dominating deviant feautre of modern students — insufficiently developed diligence and unavailability to overcoming difficulties. Development of professional and personal qualities and competences of the expert assumes updating of motivation of students, their active adaptation to educational process, increase of their responsibility for assimilation of the training program and the social behavior accompanying it. The author assume that developen negative social well-being of professional group of teachers of the higher school is determined by low prestige of this profession in society, sharp differentiation of compensation on regions, an ambiguity of the purposes of reforming of the higher school, complication and ambiguity of their professional roles. Conclusions and offers of the article are based on results of a number of the complex sociological researche, which have been carried out in 2010-2013 in higher education institutions and comprehensive schools of Ural under the scientific management and with direct participation of the author.

The spectral analysis of the Schrodinger operator on cubic lattice type graphs is developed. Similarly to the quantum mechanical tight-binding approximation, using the well known concept of the Dirichlet-to-Neumann map, asymptotic formulae for localized negative spectral bands of the Schrodinger operator on a periodic metric graph are established. The results are illustrated by numerical calculations.

We consider a risky asset following a mean-reverting stochastic process of the form We show that the (singular) diffusion equation which gives the value of a European option on S can be represented, upon expanding in Laguerre polynomials, by a tridiagonal infinite matrix. We analyse this matrix to show that the diffusion equation does indeed have a solution and truncate the matrix to give a simple, highly efficient method for the numerical calculation of the solution.

A model Laplacian with Neumann boundary conditions (Neumann problem) in a system of two cavities joined by a thin channel is investigated. An expression is obtained for the resolvent and also the first terms in the asymptotic expansions of the eigenvalues and eigenfunctions with respect to the small coupling parameter.

A class of regions in which the Laplacian for the Neumann problem has an essential spectrum is considered. The connection between the geometrical characteristics of the region and spectral properties of the Laplacian for the Neumann problem is studied in specific examples.

An explicitly solvable model of Mossbauer scattering of γ rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation Δx of the nucleus and the photon wave numberk satisfykΔx≪Eγ/EΘ, whereEγ andEΘ are typical energy levels of the photon and the oscillator states.

A new, symmetry-based approach to the selection of physically meaningful extension parameters — the deficiency subspaceN and the self-adjoint operator Γ that determines the boundary conditions — is proposed.

We reduce the spectral problem for an additively perturbed self-adjoint operator H V =H 0−V, to the dual problem of finding zeros of the operator function $$ Sign V - |V|^{1/2} [H_0 - \lambda ]^{ - 1} |V|^{1/2} , $$ and develop the Schmidt perturbation procedure for the resolvent of H v . Based on Rouche theorem for operator-valued analytic functions, we observe, in the Krein’s formula for the perturbed resolvent [H v -λI]-1, the compensation of singularities inherited from H 0, and suggest a convenient algorithm for approximate calculation of the groups of eigenfunctions and eigenvalues of the perturbed operator.

A detailed spectral analysis is carried out for the Schrodinger operator with zero-radius potentials endowed with internal structure. Explicit expressions are obtained for the resolvents and spectral projections in the case when the internal operators are compact. The corresponding scattering problem is solved.Bibliography: 22 titles.

Historically, the first general method in the spectral analysis of non-selfadjoint differential operators was the Riesz integral, complemented by the refined technique of estimating the resolvent on the contours that divide the spectrum. Using this method, Lidskij (1962) proved the summation over groups (“with brackets”) of the spectral resolution of a general regular second order differential operator. Since then, the so-called “bases with brackets” have been studied extensively by his successors (see the references in Sadovnichij (1973)). Unfortunately, the arrangement of the “brackets”, that is, the combination into one group of the sets of eigenvectors and root vectors corresponding to some neighbouring points of the spectrum, is defined non-uniquely and, to a large extent, non-constructively. Hence, as a rule, the assertions concerning bases with brackets have the character of existence theorems.

In terms of crisis economic development, one of priority goals for the technical institutes of higher education is to train engineers who are competitive on the regional labour markets. The main root of the problem of low prestige of the engineering profession in Russia, the slippage in the personnel training in technical universities lies in the depreciation of engineering work, and reduction of its social and economic attractiveness. The gap between science, education and industry leads to aging of engineering staff at the manufactures and to migration of the most talented engineers into other fields of activity. This paper analyzes current problems of engineers training organization in the Urals. The causes of sharp decline of the engineering profession social status in Russia, the fall of interest of secondary school graduates to continuation of their studies in technical institutes of higher education are reviewed. The authors show that the formation of engineering competence as a defining personal and vocational quality of a specialist involves actualization of the student's motivation, one's active and purposeful adaptation to the educational process, increasing one's responsibility for mastering the curriculum. Conclusions and suggestions of the authors are based on the results of a comprehensive sociological research conducted by them in 2011 in five high schools of the Urals (Yekaterinburg, Nizhnevartovsk and Chelyabinsk). The survey showed that during the stage of young specialists' preparation, cooperation in the «university - enterprise» system is actually shifted to the interaction, in fact, between institute of higher education and young professionals who act as the sellers of their labour. The authors believe that it makes sense to roll over (or rather - to stimulate) the active engineering work of the most productive engineering professionals who have reached retirement age. Equally sharp and critical are the issues of reproduction and preservation of professors and teachers in the technical (and not just technical) universities. In this case, the refusal of the university to use the target distribution system for its graduates should not be associated with a complete removal of the responsibility of educational institutions for the future of young engineers in the labour market.

The object of the present work is the imbedding of the spectral theory for the dissipative Schrodinger operator with absolutely continuous spectrum acting in the Hilbert space in the spectral theory of a model operator and the proof of the theorem on expansion in terms of eigenfunctions. The imbedding mentioned is achieved by constructing a selfadjoint dilation of the operator . In the so-called incoming spectral representation of this dilation the operator becomes the corresponding model operator. Next, a system of eigenfunctions of the dilation - the "radiating" eigenfunctions - is constructed. From these a canonical system of eigenfunctions for the absolutely continuous spectrum of the operator and its spectral projections are obtained by "orthogonal projection" onto .Bibliography: 22 titles.

In the geometry of total backscattering, the intensity of scattered Mossbauer γ radiation is calculated. The contribution of various scattering channels to the spectrum is discussed: resonant and nonresonant, elastic and inelastic. Scattering at samples of different thickness is considered. Only one-time scattering of γ quanta in the material is taken into account.

Conditions for separating the invariant subspaces corresponding to the absolutely continuous, discrete and singular spectrum of an abstract dissipative operator are obtained in terms of the characteristic function of the operator. The results are applied to ordinary differential operators with real potential and complex boundary conditions.Bibliography: 20 items.

We study the spectral problem $$\left. {\Delta \mathcal{U} + K^2 \mathcal{U} = 0, \frac{{\partial u}}{{\partial n}} - iK\mathcal{O}\mathcal{U}} \right|_{\partial \Omega } = 0, \mathfrak{S} \geqslant 0$$

The purpose of this article is to describe the formulation of a selfadjoint spectral problem with boundary conditions on a sufficiently thin manifold. Namely, let be a selfadjoint operator in , let be a smooth manifold, let be the restriction of to the lineal in consisting of all functions which vanish in a neighborhood of .It is shown that the deficiency elements of this restriction can be represented as "tensor layers" with densities of a definite class of smoothness, concentrated on the "boundary" of . If is sufficiently thin, there is only one family of deficiency elements, and it is analogous to the single-layer potentials. In this case, calculation of the boundary form and the description of the selfadjoint extensions appears to be quite simple. This case is studied in detail because the investigation of the simplest model of the three-particle problem of quantum mechanics reduces to it.Bibliography: 16 titles.

The scattering problem is investigated for a system of differential equations on the interval [0,a], where A is a positive matrix-valued function which jumps to I for x > a. The scattering matrix for large spectral parameter is studied, the system of resonances is described, and an expression for the resonance states in terms of the Jost solution is given. A relation is established between the resonances and the poles of the analytic continuation of the Green function. It is proved that the syrtem of resonance states corresponding to complex zeros of the scattering matrix has serial structure; namely, it splits into n Carleson series. The completeness of the system of resonances is investigated, and it is established that this system forms a Riesz basis in the corresponding space with the energy metric.Bibliography: 27 titles.

By the methods of the theory of extensions, one constructs an explicitly solvable resonator model with a small opening. One computes the high-frequency resonances, i.e., the eigenvalues of the dissipative operator, associated with the considered problem of resonance scattering in the framework of the Lax-Phillips theory.

In this article we solve the problem about the calculation of losses in a scattering problem with "Lax" and "non-Lax" channels. For the initial scattering matrix we consider the scattering matrix of the basic operator of the problem with respect to a simple unperturbed operator, which acts in a distinguished subspace (a Lax channel) that is the orthogonal sum of the incoming and outgoing subspaces. It turns out that this scattering matrix is nonunitary when the basic space contains other channels besides the distinguished one, including non-Lax channels. The concept of losses is connected with the fact that the scattering matrix is nonunitary. We calculate the losses by constructing in the orthogonal complement of a Lax channel a new selfadjoint operator, which with the original unperturbed operator forms a modified unperturbed operator. The latter has a unitary scattering matrix with respect to the basic operator of the problem. We explain the significance of the elements of the new scattering matrix that include the original matrix.Bibliography: 9 titles.

CONTENTSIntroduction § 1. Boundary values for abstract operators and the Krein formula for generalized resolvents § 2. Zero-radius potentials in diffraction problems § 3. Quantum-mechanical problems with energy-dependent potentials Conclusion References