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In the research presented in the monograph, a theoretical analysis of the essence, nature and genesis of patriotic education is carried out, the content, role and features of patriotic education in society are revealed, state and regional policy as the foundation and means of patriotic education is studied, priority directions of state and regional policy on patriotic education are determined, alternative functional models of patriotic education are considered, an integrative model of patriotic education is conceptually profiled and empirically specified. The book is intended for managers and employees of state and municipal administration bodies, specialists and experts in patriotic education. It is also addressed to teachers of higher and secondary specialized educational institutions, researchers of academic institutes and research centers. It will be useful for students of higher educational institutions studying in social and humanitarian specialties.

This paper discusses resonance effects to advance a classical earthquake model, namely the celebrated M8 global test algorithm. This algorithm gives high confidence levels for prediction of Time Intervals of Increased Probability (TIP) of an earthquake. It is based on observation that almost 80% of earthquakes occur due to the stress accumulated from previous earthquakes at the location and stored in form of displacements against gravity and static elastic deformations of the plates. Nevertheless the M8 global test algorithm fails to predict some powerful earthquakes. In this paper we suggest the additional possibility of considering the dynamical storage of the elastic energy on the tectonic plates due to resonance beats of seismogravitational oscillations (SGO) modes of the plates. We make sure that the tangential compression in the middle plane of an “active zone” of a tectonic plate may tune its SGO modes to the resonance condition of coincidence the frequencies of the corresponding localized modes with the delocalized SGO modes of the complement. We also consider the beats arising between the modes under a small perturbations of the plates, and, assuming that the discord between the perturbed and unperturbed resonance modes is strongly dominated by the discord between the non-resonance modes estimate the energy transfer coefficient.

Spin-depending scattering is studied on a two-dimensional quantum network constructed of a quantum well with three semi-infinite quantum wires (an input wire and two terminals ) attached to it. The spin-orbital interaction causing selective scattering in the network is described by the Rashba Hamiltonian. The transmission of electrons across the well from the input quantum wire to terminals is caused by the excitation of a resonance oscillatory spin-mode in the well. For thin quantum networks we suggest also an approximate formula which defines the resonance transmission of electrons across the quantum well, depending on the shape of the standing spin-mode in the quantum well.

We present a brief critical review of modern theoretical interpretations of the low-threshold field emission phenomenon for metallic electrodes covered with carbon structures, taking the latest experiments into consideration, and confirming the continuity of spectrum of resonance states localized on the interface of the metallic body of the cathode and the carbon cover. Our proposal allowed us to interpret the double maxima of the emitted electron’s distribution on full energy. The theoretical interpretation is presented in a previous paper which describes the (1 + 1) model of a periodic 1D continuous interface. The overlapping of the double maxima may be interpreted taking into account a 2D superlattice periodic structure of the metal-vacuum interface, while the energy of emitted electrons lies on the overlapping spectral gaps of the interface 2D periodic lattice.

It was shown earlier that the excitation threshold of electron emission from carbon nanoclusters is extremely low (two to three orders of magnitude lower than for metals and semiconductors). In this work, this effect is confirmed by direct experiments with graphene and graphene-like structures. A new model explaining this effect by resonance tunneling due to size quantization is suggested. A low-threshold explosive emission is discovered.

A Helmholtz resonator is a shell Ωshell separating a compact cavity Ωint from a noncompact outer domain Ωout. A small opening Ωδ in the shell connects the cavity with the outer domain, causing the transformation of real eigenfrequencies of the Neumann Laplacian in the cavity into the complex scattering frequencies of the full spectral problem for the Neumann Laplacian on Ω = ℝ3\Ωshell.

The Neumann Schrodinger operator \( \mathcal{L} \) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ωint and a few semi-infinite straight leads ωm, m = 1, 2, ..., M, of width δ, δ ≪ diam Ωint, attached to Ωint at Γ ⊂ ∂Ωint. The potential of the Schrodinger operator lω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrodinger operator Lint on Ωint embedded into the open spectral branches of lω with oscillating solutions χ±(x, p) = \( e^{ \pm iK_ + x} e_m \) of lωχ± = p2χ±. The exponent of the open channels in the wires is $$ K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $$ , with constant em, on a relatively small essential spectral interval Δ ⊂ [0, π2δ−2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping $$ \mathcal{N} = \frac{{\partial P_ + \Psi }} {{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $$ as $$ S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $$ . We derive an approximate formula for \( \mathcal{N} \) in terms of the Neumann-to-Dirichlet mapping \( \mathcal{N}_{\operatorname{int} } \) of Lint and the exponent K− of the closed channels of lω. If there is only one simple eigenvalue λ0 ∈ Δ, Lintφ0 = λ0φ0 then, for a thin junction, \( \mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with $$ \vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma ) $$ and \( P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \), $$ S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }} {{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda ) $$ . The related boundary condition for the components P+Ψ(0) and P+Ψ′(0) of the scattering Ansatz in the open channel \( P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M ) \) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where $$ \frac{{\bar \Psi _m }} {{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }} {{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }} {{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M $$ (1) , $$ \sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0 $$ (1) . Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrodinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

Carbon nano-cluster cathodes exhibit a low threshold electron emission, which is 2-3 orders lower than on metals and semiconductors. We confirm the effect by direct experiments with graphene structures. We are suggesting a model based on the interference electrons wave function in 3D-space charge region of carbon structure interface with vacuum. The low-threshold emission is explained, in frames of the model, by the resonance properties of the barrier formed on the interface. Also in the following topics: interpretation of recent experimental findings for saturation of the field emission; local spectral analysis of multidimensional periodic lattices: dispersion via DN-map; examples of iso-energetic surfaces associated with solvable models of periodic lattice; Lagrangian version of the operator extension algorithm; solvable models of selected one-body spectral problems; quantum dot attached to the node of a quantum graph; a solvable model of a discrete lattice and spectral structure of a 1D superlattice via analytic perturbation procedure.

Plasma waves in two-dimensional electron channels with nontrivial geometry and governing electrodes are described by the hydrodynamical equations combined with the Poisson equation for the self-consistent electric potential. If the amplitudes of the oscillations of the velocity, the concentration, and the potential are much smaller than the stationary values of these variables, then the hydrodynamical equation for oscillations can be linearized and transformed to the wave equation on a two-dimensional network. We consider the scattering problem for the wave equation and develop a semi-analytic method to calculate the transmission coefficients through the junction. The formulas for the transmission coefficients have resonance character and can be used in design of manipulated 2D-networks of plasma channels.

We consider the spectral problem for the Schrodinger operator ℒ on a quantum network, represented as a union of a finite number of quantum wells and finite or semi-infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrodinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi-infinite wires. We split the Schrodinger operator, depending on the Fermi level E, into an orthogonal sum of two operators ℒ lΛ ⊕ LΛ, which have the complementary branches of the absolutely continuous spectra coincident, counting multiplicity, respectively with the unions Σ = ∪σopen, Σ = ∪σclosed of absolutely continuous branches σ = [λσ, ∞) of open and closed channels in the semi-infinite wires, with thresholds λσ Λ, respectively. This enables us to reduce the problem of evaluation of the scattering matrix on the network to the solution of a finite linear system, involving the Dirichlet-to-Neumann map of the intermediate Hamiltonian LΛ. We calculate the intermediate Dirichlet-to-Neumann map approximately, for thin networks, based on the classical DN-map of the relevant Schrodinger operator on the compact part of the network. To this end we develop a modified analytic perturbation procedure for meromorphic operatorfunctions with small one-dimensional residues. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

One-electron spin-dependent scattering problem is considered on the star-shaped 2-d quantum network consisting of a quantum well and few relatively thin leads attached to it. The resonance nature of transmission coefficients is revealed based on accurate analysis of interaction between the running spin-waves in the leads and the standing spin-waves in the quantum well. Modeling of the transport problem on the two-dimensional junction by the 1-d scattering problem on the corresponding quantum graph is discussed.

One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.

The article discusses current problems of the rural youth migration to the cities in the context of socio-residential assimilation, suggesting a convergence of lifestyles of different large and small social groups living in different types of settlements. The author analyzes the significance of social environments for the formation of political culture of young villagers and the development of their values in the sphere of self-identification. The article suggests that successful strategic socio-economic development (prosperity) of large cities, especially megacities, should be closely connected with their responsibility for the adequate development of small towns and rural settlements in order to reproduce and save labor potential of agricultural production in the region and ensure decent life for all actors in this production. The authors’ reflections and conclusions are based on the results of comparative sociological surveys of the urban and rural youth, carried out in various cities and rural settlements of some RF subjects within the Ural Federal District in 2008–2014

We consider the spectral problem for the adjacency matrix of a discrete star-graph composed of a compact cluster and a few semi-infinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop the Lax–Phillips scattering theory for the corresponding discrete wave equation.

We suggest an elementary example of a simplest starlike graph that shows that the analytical disjunction may occur not on only at isolated points of the spectral parameter, but also on whole intervals. Bibliography: 1 title.

For the Friedrichs model, obtained as a one-dimensional ?-perturbation of the orthogonal sum of the momentum and a finite Hermitian matrix: , the scattering matrix is presented as S?(p) = [I + i?M(p)]?1[I ? i?M(p)]. The rational Nevanlinna-class Krein?Weyl function M is associated with the operator A and has poles at the eigenvalues of A. It is proven that for any eigenvalue ?0 of A there exists an intermediate operator , which is constructed as a one-dimensional perturbation of the orthogonal sum of the momentum and an appropriate one-dimensional operator?a solvable model, which plays the role of an intermediate operator in the scattering problem to the pair . The scattering matrix S0? to the pair is an analytic function of ? and the total scattering matrix to the pair can be factorized as a product where S?0 is the scattering matrix to the pair . It is represented by a single Blaschke factor with a pole and zero approaching ?0 when ? ? 0. The non-analytic factor describes creation of the resonance from the eigenvalue ?0 of the operator A.

We propose iteration methods for solving the Dirichlet problem in domains with involved geometry. Such problems arise in relation to the problem of optimizing quantum dot and antidot infrared detectors. We estimate the deviation of an approximate solution from the exact solution.

It had been observed by Linkov, Petrova and Osipov (1992) that there exist periodic 4-6 hours pulses of 200 microHz seismogravitational oscillations ( SGO ) before 95 % of powerful earthquakes. We explain this by beating between an oscillation eigenmode of a whole tectonic plate and a local eigenmode of an active zone. The beating transfers the oscillation energy from the remote zone of the tectonic plate to the active zone, triggering the earthquake. Oscillation frequencies of the plate and ones of the active zone are tuned to a resonance by an additional compression applied to the active zone due to collision of neighboring plates or the magma flow in the liquid underlay of the astenosphere ( the upper mantle). In the case when there are three or more SGO with incommensurable difference frequencies the SGO beating pattern looks quasi-random, thus masking the non-random nature of the beating process. Nevertheless, we are able to discuss a possibility of the short term earthquakes predictions based on an accurate monitoring of the beating dynamics., Comment: Journal of Seismology (JOSE) DOI 10.1007/s10950-015-9525-9

A Scattering problem is studied for Neumann Laplacian with a continuous potential on a domain with a smooth boundary and few semi-infinite wires attached to it at the points of contact on the boundary. In resonance case when the frequency of the incoming waves in the wires coincides with some resonance frequency of the domain the approximate formula for the transmission coefficient from one wire to another is derived: in the case of weak interaction between the domain and the wires the transmission coefficient is proportional to the product of values of the corresponding resonance eigenfunction of inner problem at the corresponding points of contact.

Narrow-gap semiconductors, used in quantum network engineering, are characterized by small effective electron masses on the Fermi level and hence by high electron mobility in the lattice. We construct an explicitly solvable model that clarifies one possible mechanism for small effective masses to appear. Another mathematical model constructed here describes a possible mechanism for using a traveling wave to control an alternating quantum current in a one-dimensional lattice.

Is there any hope for quantum computing to challenge the Turing barrier, i.e., to solve an undecidable problem, to compute an uncomputable function? According to Feynman's '82 argument, the answer is negative. This paper re-opens the case: we will discuss solutions to a few simple problems which suggest that quantum computing is theoretically capable of computing uncomputable functions. Turing proved that there is no “halting (Turing) machine” capable of distinguishing between halting and non-halting programs (undecidability of the Halting Problem). Halting programs can be recognized by simply running them; the main difficulty is to detect non-halting programs. In this paper a mathematical quantum “device” (with sensitivity e) is constructed to solve the Halting Problem. The “device” works on a randomly chosen test-vector for T units of time. If the “device” produces a click, then the program halts. If it does not produce a click, then either the program does not halt or the test-vector has been chosen from an undistinguishable set of vectors Fe, T. The last case is not dangerous as our main result proves: the Wiener measure of Fe, T constructively tends to zero when T tends to infinity. The “device”, working in time T, appropriately computed, will determine with a pre-established precision whether an arbitrary program halts or not. Building the “halting machine” is mathematically possible. To construct our “device” we use the quadratic form of an iterated map (encoding the whole data in an infinite superposition) acting on randomly chosen vectors viewed as special trajectories of two Markov processes working in two different scales of time. The evolution is described by an unbounded, exponentially growing semigroup; finally a single measurement produces the result. PACS: 03.67.Lx

We consider quantum wave propagation in one-dimensional quasiperiodic lattices. We propose an iterative construction of quasiperiodic potentials from sequences of potentials with increasing spatial period. At each finite iteration step, the eigenstates reflect the properties of the limiting quasiperiodic potential properties up to a controlled maximum system size. We then observe approximate Metal–Insulator Transitions (MIT) at the finite iteration steps. We also report evidence on mobility edges, which are at variance to the celebrated Aubry–André model. The dynamics near the MIT shows a critical slowing down of the ballistic group velocity in the metallic phase, similar to the divergence of the localization length in the insulating phase.

The problem of construction of quantum multiplexer is discussed. A possible construction based on resonance transport properties of quantum waveguides coupled through small windows is considered. Small apertures play double role of “connecting channels" and “resonant elements". Transmission coefficients for the system are determined. The workability of the device as a quantum switch to one of three (or to two of three) channels is discussed. Control parameters for the switch are electron energy and bias voltage.

The problem of constructing a quantum multiplexer is discussed. A possible construction based on the properties of quantum waveguides connected with a quantum resonator is considered. A descriptive model of ballistic transport in 1D-2D-1D system is suggested. It is analogous to the zero-range potential method and based on the theory of self-adjoint extensions of symmetric operators. A general formula for transmission coefficient for the system, particularly in the case of presence of a magnetic field, is obtained. A concrete realization of the formula for the quantum system in question is developed and analyzed. Quantum resonators with two and four leads are studied. A possible application of the result for constructing a quantum multiplexer is discussed. Control parameters for the device are electron energy (momentum filter) or bias voltage related with the radius of the quantum resonator (multiplexer). If there is a magnetic field then a value of a magnetic field B is an additional control parameter.

A model of the periodic array of quantum antidots in the presence of a uniform magnetic field is suggested. The model can be conceived as a periodic lattice of resonators(curvilinear triangles)connected through ‘‘infinitely small’’ openings at the vertices of the triangles. The model Hamiltonian is obtained by means of operator extension theory in indefinite metric spaces. In the case of rational magnetic flux through an elementary cell of the lattice, the dispersion equation is found in an explicit form with the help of harmonic analysis on the magnetic translation group. It is proved, at least in the case of integer flux, that the spectrum of the model Hamiltonian consists of three parts: (1) Landau levels (they correspond to the classical orbits lying between antidots); (2) extended states that correspond to the classical propagation trajectories; and (3) bound states satisfying the dispersion equation; they correspond to the classical chaotic orbits rotating around single antidots. Among other things, methods of finding the Green’s function for some planar domains with curvilinear boundaries are derived.

A method of deducing an expansion theorem in resonance states for the two-dimensional Friedrichs model\(A = \tfrac{1}{{2i}}(U_\varphi - U_\varphi ^ + )\) is suggested. The construction is based on detailed Lax-Phillips analogs of the corresponding unitary Friedrichs model\(U_\varphi = z(1 + \tfrac{{2i\alpha }}{{1 - i\alpha }}P_\varphi )\), perturbed by the projector Pϕ, generated by an element ϕ possessing special analytic properties. Bibliography: 11 titles.

The two‐body system on a graph with one junction is considered. The effective three‐body scattering problem turns out to be exactly solvable for pointwise interactions. Additional degrees of freedom corresponding to a dynamics of some structure (e.g., an atomic cluster) located in the junction (point of common contact) of three thin electrodes are considered. These degrees of freedom bring effective energy‐dependent interaction into the effective Schrodinger equation in the scattering channel. The wave function of the system is constructed in the explicit form using the extension theory methods. The obtained results are applied to the qualitative description of a simple three‐electrode nanoelectronic device. The perturbation theory approach based on the analysis of the Liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function.

Carbon nano-cluster cathodes exhibit a low threshold electron emission, which is 2–3 orders lower than on metals and semiconductors. We confirm the effect by direct experiments with graphene structures. We are suggesting the model based on interference electrons wave function in 2D-space charge region of carbon and Tamm's surface states of dangling bonds. This barrier is transparent for electrons at Fermi level due to the resonance.

First of all: disregard all of so called "preliminary physical expectations", because all these expectations are just a pseudonym for prejudices elaborated by the older generation. These words belong to Dirac, not to me. The right way of creating new physics is different: one should begin with a beautiful mathematical idea. But it should be really beautiful! No special relations to physics is compulsory. But if it is really beautiful, it will certainly match useful physical applications, though it is not predefined, what sort of applications and where: it depends on physical consequenceswhich may be extracted from the mathematical scheme.

Two exactly solvable models illustrating the links between spectral properties of Hamiltonians, connections on the induced Hilbert bundles and topological characteristics of the basis spaces are considered. The first of them is based on the extension theory for symmetric operators and the second on the one-dimensional Laplace operator with parametrical boundary conditions.

A solvable model of the scattering of a one‐dimensional particle by the familiar ‘‘middle‐third’’ Cantor set is considered. It is shown that the presence of positive energy levels in such a model is typical. From the high‐energy behavior of the scattering, data computation of the Hausdorff dimension of the scatterer is suggested.

The asymptotic estimations, as period tends to infinity, for negative spectral bands of three‐dimensional periodical Schrodinger operator who’s potential was generated by shifting along the cubic lattice of a negative function with compact support are described herein. The problem is of primary importance for solid state physics.

The relations between the Schrodinger and wave evolutions are investigated. A generalization of the Lax-Phillips scattering theory for the case of the Schrodinger equation is constructed. An analog of the wave decay operator is obtained. The general ideas are illustrated by an explicitly solvable model of multichannel scattering on the halfaxis. A Lax-Phillips scattering theory is developed for the model operator obtained by a self-adjoint perturbation of the background operator. © 1993 John Wiley & Sons, Int.

Detonation carbon materials (DCM) composed of non-equilibrium nano-structures show the low-threshold field emission (LTFE). These materials have forward-looking application especially due to high reproducibility of the LTFE-phenomenon on a surface of emitter, where the emitting centers are homogeneously distributed. In this paper we link the effect of LTFE to the nature of the corresponding wave functions based on the experiment results obtained for DCM by the field effect on electrolytes. As we had shown before DCM had been described by an ultra-relativistic dispersion function with extremely small effective mass of electrons and the size-quantization effect had been observed in DCM at even room temperature. Our results based on emission and electrolyte technics of the field-effect measurements in DCM along with modern observations of the field emission in strong electric fields allowed to propose a new resonance transmission model for LTFE-phenomenon, which is alternative to most widely discussed models based on the field-enhancing factors or barrier-lowering mechanisms.

A model of the Markovian chain of N atoms is constructed with the help of extension theory. Behavior of averaged wave functions of random Dirac operator is studied for finite and large N with respect to parameters of the model and Markovian process. Study of a random Dirac operator is connected with the study of nonrandom double‐dimensional non‐self‐adjoint Dirac operator. Dynamics of bands of effective periodic operator with respect to the Markovian parameter is presented.

The single-electron transport through a junction of quantum network is modeled as a specific scattering problem for the Schrodinger operator on the system of semi-infinite cylindrical domains (quantum wires) short-circuited by a compact domain (quantum well or dot). For calculation of the one-body scattering parameters of any junction having form of a compact domain with piece-wise smooth boundary and attached thin wires a semi-analytic perturbation procedure based on a specially selected intrinsic large parameter is suggested. The approximate scattering matrix of a thin junction obtained in this way is the scattering matrix of the corresponding solvable model.

A local scattering theory on finite spectral interval for pairs of selfadjoint operators, which are different extensions of the same densely definite symmetric operator, is developed. The present investigation was motivated by scattering problems for Schrodinger operators modeling the energy operator of electrons in real quantum networks.

We consider a one-body spin-less electron spectral problem for a resonance scattering system constructed of a quantum well weakly connected to a noncompact exterior reservoir, where the electron is free. The simplest kind of the resonance scattering system is a quantum network, with the reservoir composed of few disjoint cylindrical quantum wires, and the Schrodinger equation on the network, with the real bounded potential on the wells and constant potential on the wires. We propose a Dirichlet-to-Neumann-based analysis to reveal the resonance nature of conductance across the star-shaped element of the network (a junction), derive an approximate formula for the scattering matrix of the junction, construct a fitted zero-range solvable model of the junction and interpret a phenomenological parameter arising in Datta- Das Sarma boundary condition, see [14], for T-junctions. We also propose using of the fitted zero-range solvable model as the first step in a modified analytic perturbation procedure of calculation of the corresponding scattering matrix.

We suggest a method of fitting of a zero-range model of a tectonic plate under a boundary stress on the basis of comparison of the theoretical formulae for the corresponding eigenfunctions/eigenvalues with the results extraction under monitoring, in the remote zone, of non-random (regular) oscillations of the Earth with periods 0.2-6 hours, on the background seismic process, in case of low seismic activity. Observations of changes of the characteristics of the oscillations (frequency, amplitude and polarization) in course of time, together with the theoretical analysis of the fitted model, would enable us to localize the stressed zone on the boundary of the plate and estimate the risk of a powerful earthquake at the zone., 24 pages, 4 figures

A general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy‐dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

Plasma waves in a slot-diod with governing electrodes are described by the linearized hydrodynamic equations. Separation of variables in the corresponding scattering problem is generally impossible. Under natural physical assumption we reduce the problem to the second order differential equation on the slot with an operator weight, defined by the Dirichlet-to-Neumann map of the three-dimensional Laplacian on the complement of the electrodes and the slot. The reduction is based on a formula for the Poisson map for the exterior Laplace Dirichlet problem on the complement of a few standard bodies in terms of the Poisson maps on the complement of each standard body.