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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.

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.

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.

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

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.

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.

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.

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.

The quantum graph plays the role of a solvable model for a two-dimensional network. Here fitting parameters of the quantum graph for modelling the junction is discussed, using previous results of the second author., Replaces unpublished draft on related research

The mathematical design of a realistic three-position quantum switch controlled by the classical electric field is suggested in form of a circular quantum well — a unit disc on a plane with four straight channels attached to it. This device implements a triple splitting of an input waveguide. The magnitude of the constant electric field directed parallel to the disc may be defined such that rotation of this field in the plane of the device permits manipulation of the electron current through the triple splitting. The problem of calculating of the current through the switch is reduced to the construction of scattered waves for the Schrodinger operator on the corresponding composite domain with the homogeneous Dirichlet conditions on the boundary. The Dirichlet boundary conditions are found to correspond most closely to real experimental conditions on the boundary of a deep quantum well. Explicit expression for transmission coefficient from one channel to another is obtained.

A solvable model of the three-positions quantum switch — triadic relay — is constructed as a circular domain with 4 one-dimensional wires attached to it. In resonance case, when Fermi level in the wires coincides with some energy level in the domain, the magnitude of the governing constant electric field is specified such that manipulation of the quantum current through the relay may be done via the change of the direction of the field in a plane parallel to the circle.

The mathematical model of a simplest quasi-one-dimensional quantum network constructed of relatively narrow waveguides (the width of the waveguide is less than the de Broghlie wavelength of the electron in the material) is developed. This model allows to reduce the problem of calculating the current through the quantum network to the construction of scattered waves for some Schrodinger equation on the corresponding one-dimensional graph. We consider a graph consisting of a compact part and few semiinfinite rays attached to it via some boundary condi-tion depending on a parameter β (analog of the inverse exponential “mass” e -bH of a potential barrier H separating the rays from the compact part of the graph). This parameter regulates the connection between the rays and the compact part. Spectral properties of the Schrodinger operator on this graph are described with a special emphasis on the resonance case when the Fermi level in the rays coincides with one of eigenvalues of the nonperturbed Schrodinger operator on the ring. An explicit expression is obtained for the scattering matrix in the resonance case for weakening connection β → 0 between the rays and the compact part.

Finite rank generalized singular perturbations of selfadjoint operators are studied. It is proved that the restriction of the arising operator relation to the subspace generated by the original Hilbert space is a selfadjoint operator (if the perturbation is singular). The relations with the corresponding scattering problem are investigated. In particular, the case where the scattering matrix has physical behavior at infinity is examined.

A model of the scattering of a structureless pointlike particle on a spherical bag with an internal structure, imitating ‘‘quark’’ degrees of freedom, is considered. It is assumed that the bag is a dynamical object and its radius plays the role of additional dynamical variable. The energy of the collision is distributed among the quark excitations and the vibration degrees of freedom of the bag surface. In the frame of the theory of extensions the Hamiltonian of the coupled bag‐quark system interacting with the pointlike particle is constructed. The formal multichannel S matrix of the problem is obtained.

A Multichannel Schrodinger equation with a periodic matrix potential is investi gated. Such an equation arises during the reduction of the multidimensional Schrodinger equation periodic in one direction to the set of connected one-dimensional equations. It is related to the calculations of the electron spectrum in solid state, superlattices and low dimensional lattices. Periodic structure of the multichannel Hamiltonian is con sidered as infinite number of the periodically situated short range perturbations of the translation invariant matrix operator. Relations between the spectral properties of the single scattering centers and the band structure of the periodic problem are investigated. The first section is devoted to the finite dimensional Schrodinger equation. Relations between the scattering on a finite crystal and the corresponding periodic problem are clarified. It is shown that the bound state energies of a single center define bands of the continuous spectrum, resonances produce spectral gaps in the spectrum. The second section is devoted to the scattering on a one-dimensional chain in the three-dimensional space, which corresponds to the infinite dimension of the matrix potential. An exactly solvable model is constructed with the help of zero range potentials.

Exactly solvable models are very important in modern statistical physics. Such models can give essential information about investigating physical objects. Unfortu nately these models can not posses all properties of real systems. But exact analytical solutions permit us to calculate very important quantities, which can not be done for the real ones, or can be done only numerically. One of the most popular models in statistical mechanics is the model of hard core spheres. This model has been used to in vestigate properties of the second ([8,9]) and third ([10]) virial coefficients. Even phase transitions can be observed in this model [7].

G помощью теории расширений построена нетривиальная матрица рассеяния для системы трех одномерных частиц в модели граничных условий. Исследована аналитическая структура построенной ^-матрицы на энергетической поверхности.

Методом теории расширений нерелятивистского гамильтониана с выходом в дополнительное пространство внутренних степеней свободы построена модель бинарных реакций в системе частиц, обладающих нетривиальной внутренней структурой. Модель применяется для описания адрон-адронного рассеяния при низких и промежуточных энергиях.

A spectral analysis is made of a Schrodinger operator with zero-range potential of a one- or two-dimensional lattice in the resonances of an isolated atom and the spectral properties of the crystal is established

The methods of the theory of extensions to an auxiliary Hilbert space are used to construct in the two-particle sector a scattering theory for particles possessing internal structure. The analytic properties of the amplitudes of resonance scattering and Green's functions corresponding to a class of singular energy-dependent interactions are investigated.

We consider the quantum Lorentz gas (QLG) in the plane. We show that the spectral properties of the quantum Lorentz gas can be obtained from the study of a homogeneous system of one-dimensional integral equations. The qualitative spectral analysis is performed, and the spectrum is shown to have a band structure. The wave functions in the complete configuration space can be constructed in terms of the solutions of the obtained effective equations. We apply the obtained result to a simpler Lorentz gas model including additional symmetry.