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Practically relevant problems of quadratic optimization often contain multidimensional arrays of variables interconnected by linear constraints, such as equalities and inequalities. The values of each variable depend on its specific meaning and can be binary, integer, discrete, and continuous. These circumstances make it technically difficult to reduce the original problem statement to the QUBO form. The paper identifies and considers three main transformations of the original problem statement, namely, the transition from a multidimensional to a one-dimensional array of variables, the transition in mixed problems to binary variables, and the inclusion of linear constraints in the objective function in the form of quadratic penalties. Convenient formulas for calculations are presented and proven, simplifying the implementation of these transformations. In particular, the formulas for the transition in the problem statement from a multidimensional to a one-dimensional array of variables are based on the use of the Kronecker product of matrices. The considered transformations are illustrated by numerous examples., Comment: 26 pages, 3 figures; in Russian

In this paper the detailed investigation of one of the most interesting models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is continued; the starting point of dynamic analysis for potentials, wave functions and energy spectrum was carried on by the same authors in the paper. The generalization of these results on the case of nonzero temperature includes the approximate calculation of the partition function of the model and then also the Massieu Planck thermodynamic potential and its most important derivatives, i.e. internal energy, heat capacity and pressure. The analysis of the results is carried on separately for low and high temperature regions because of the lack of exact unifying expressions of the potential at all temperatures., Comment: 13 pages, in Russian

In a two-dimensional approximation, the probability density and current for a photoelectron near the localization of a quantum vortex are theoretically investigated. The wave function in the momentum representation, which we found earlier, is simplified near zero, corresponding to the center of the vortex. This allows us to obtain a simple analytical expression for it, which has the structure of a Gaussian wave packet and contains basic information about the vortex. It is used to analyze the temporal evolution of a quantum vortex, both in momentum and coordinate space. The effect of the intensity of an ionizing ultrashort laser pulse on the shape of a quantum vortex is also investigated., Comment: in Russian language

In the paper a model of a single-atom laser with incoherent pumping is theoretically investigated. In the stationary case, a linear homogeneous differential equation for the phase-averaged Hussimi Q-function is derived from the equation for the density operator of the system. In the regime when the coupling of the field with an atom is many times stronger than the coupling of the field with the reservoir providing its decay, an asymptotic solution of this equation is found. This solution makes it possible to describe some statistical features of a single-atom laser, in particular the weak sub-Poissonian photon statistics., Comment: in Russian, to be published in JETP

We consider various iterative algorithms for solving the linear equation $ax=b$ using a quantum computer operating on the principle of quantum annealing. Assuming that the computer's output is described by the Boltzmann distribution, it is shown under which conditions the equation-solving algorithms converge, and an estimate of their convergence rate is provided. The application of this approach to algorithms using both an infinite number of qubits and a small number of qubits is discussed., Comment: In Russian, 18 pages, 2 figures

The quantum vortices formed as a result of barrier-suppression ionization of a two-dimensional hydrogen atom by an ultrashort laser pulse are theoretically investigated. Using an analytical expression for the wave function of a photoelectron in the momentum representation, the probability flux density is investigated. In this case, both the standard definition of a flux and an alternative one are used. The latter, due to the sensitivity to the phase of the wave function, makes it possible to identify quantum vortices in the momentum space., Comment: in Russian language, to be published in special issues of journals published by the Ioffe Institute (extended abstracts of the Physics A.SPb/2023 conference reports)

In this paper the detailed investigation of one of the most interested models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is presented. This model includes as particular cases two most popular and valuable models: the quasi free particle in the box with impenetrable hard walls (i.e., the model with confinement) and Bloch quantum harmonic oscillator, which is unconfined in space; both models are frequently and effectively exploited in modern nanotechnology e.g., in quantum dots and magnetic traps. We give the extensive and elementary exposition of the potentials, wave functions and energetic spectra of all these interconnected models. Moreover, the pressure operator is defined following the lines of G. Helmann and R. Feynman which were the first who introduced this idea in the late 30ies in quantum chemistry. By these means the baroenergetic equation of state is obtained and analyzed for all three models; in particular, it is shown the absence of the pressure for the Bloch oscillator due to the infinite width of the box. The generalization of these results on the case of nonzero temperature will be given later., Comment: in Russian

This paper is a natural continuation of our previous paper arXiv:1011.4173 . We illustrated earlier that in classical Hamilton mechanics, for overwhelming majority of real chaotic macroscopic systems, alignment of their thermodynamic time arrows occurs because of their low interaction. This fact and impossibility to observe entropy decrease at introspection explain the second law of thermodynamics. The situation in quantum mechanics is even a little bit easier: all closed systems of finite volume are periodic or nearly periodic. The proof in quantum mechanics is in many respects similar to the proof in classical Hamilton mechanics - it also uses small interaction between subsystems and impossibility to observe entropy decrease at introspection. However, there are special cases which were not found in the classical mechanics. In these cases one microstate corresponds to a set of possible macrostates (more precisely, their quantum superposition). Consideration of this property with use of decoherence theory and taking into account thermodynamic time arrows will introduce new outcomes in quantum mechanics. It allows to resolve basic paradoxes of quantum mechanics: (a) to explain the paradox of wave packet reduction at measurements when an observer is included in the system (introspection) (paradox of the Schrodinger cat); (b) to explain unobservability of superposition of macroscopic states by an external observer in real experiments (paradox of Wigner's friend); (c) to prove full equivalence of multi-world and Copenhagen interpretations of quantum mechanics; (d) to explain deviations from the exponential law at decay of particles and pass from one energy level to another (paradox of a kettle which will never begin to boil)., Comment: 42 pages in Enlish and in Russian

We give a new elementary proof of the theorem by Ambainis et al. that for a quantum walk, the probability amplitudes of absorption at the initial point after 4n steps are proportional to the Catalan numbers. We also calculate the absorption probabilities at points close to the initial one for the first time., Comment: in Russian

One of the most interesting models of the nonrelativistic quantum mechanics (in one-dimensional case) introduced by G. Poeschl and E.Teller at 1933 is considered. It is shown that this model possesses as its limiting cases two most popular quantum models: the quasi-free quantum particle in the box as well as the quantum harmonic oscillator introduced by F. Bloch at 1932., Comment: 6 pages, Russian, New technologies of science, technology, pedagogy of higher education, International scientific-practical conference "Science-society of technology-2017"

The paper examines the prominent algorithm D-MORPH to search for the optimal control of a quantum system in order to implement desired unitary evolution of the quantum system at the final time, and reveals new mathematical expressions for various orders' corrections to the algorithm, that include information about the commutators of the system's Hamiltonian. Inclusion of such corrections results in faster optimal quantum control's search with high precision, i.e. allows saving of computational resources., Comment: This is an English translation of the article originally published in Russian

A system of interacting classical oscillators is discussed, similar to a quantum mechanical system of a discrete energy level, interacting with the energy quasi-continuum of states considered Fano. The limit of a continuous spectrum is analyzed together with the possible connection of the problem under study with the generation of coherent phonons., Comment: In Russian

The aura of mystery surrounding quantum physics makes it difficult to advance quantum technologies. Demystification requires methodological techniques that explain the basics of quantum technologies without metaphors and abstract mathematics. The article provides an example of such an explanation for the BB84 quantum key distribution protocol based on phase coding. This allows you to seamlessly get acquainted with the real cryptographic installation QRate, used at the WorldSkills competition in the competence of "Quantum Technologies"., Comment: in Russian

This paper describes the Lorentz two-spinors proposing to use them instead of Dirac four-spinors and quaternions., Comment: in Russian, in English

We consider nonlinear spectroscopic effects - interaction-enhanced double resonance and spectrum instability - that appear in ultracold quantum gases owing to collisional frequency shift of atomic transitions and, consequently, due to the dependence of the frequencies on the population of various internal states of the particles. Special emphasis is put to two simplest cases, (a) the gas of two-level atoms and (b) double resonance in a gas of three-level bosons, in which the probe transition frequency remains constant., Comment: 7 pages, 4 figures; Proceedings of the XXXVI Conference on Low Temperature Physics NT-36 (in Russian), July 2-6, 2012, Ioffee Physicotechnical Institute, St. Petersburg, Russia; accepted for publication in JETP. This paper draws partly from arXiv:1101.2559

It is shown that the absence of an objective existence of the results of quantum measurements cannot be proved by known experiments. There are also general arguments confirming this conclusion., Comment: 3 pages, in Russian, 0 figures

A theoretical model of endogenous fluctuations of the norm of the wave function, consistent with the standard quantum theory, is presented. These fluctuations are a subsystem of endogenous quantum fluctuations and describe one of the decoherence channels and the dynamics of the collapse of the quantum state., Comment: 26 pages, 0 figures. To be published in the Proceedings of the XXIV Annual Conference of the Institute of Chemical Physics RAS. Section "Dynamics of chemical and biological processes". 2019 [in Russian]. In second version: 1. Corrected references to formulas (27) on page 15 and (28) on page 17. 2. Last paragraph added indicating the state program. 3. Original Russian text added

The certainty principle (2005) allowed to conceptualize from the more fundamental grounds both the Heisenberg uncertainty principle (1927) and the Mandelshtam-Tamm relation (1945). In this review I give detailed explanation and discussion of the certainty principle, oriented to all physicists, both theorists and experimenters., Comment: One paper, in Russian and in English, no figures

A more detailed derivation of the Heisenberg uncertainty principle from the certainty principle is given., Comment: One paper, in Russian and in English, no figures

In the paper we study the discrete spectrum of a pair of quantum two-dimensional waveguides having common boundary in which a window of finite length is cut out. We study the phenomenon of new eigenvalues emerging from the threshold of the essential spectrum when the length of window passes through critical values. We construct the asymptotics expansions for the emerging eigenvalues with respect to small parameter which is the difference between current length of the window and the nearest critical value. We also study the behaviour of the spectrum when the length of the window increases unboundedly and construct asymptotics expansions with respect to great parameter which is a length of the window., Comment: The original text was written in Russian. This is a translation made by the author. In the revised version just some minor changes were made

The notion of the quantum angle is introduced. The quantum angle turns out to be a metric on the set of physical states of a quantum system. Its kinematics and dynamics is studied. The certainty principle for quantum systems is formulated and proved. It turns out that the certainty principle is closely connected with the Heisenberg uncertainty principle (it presents, in some sense, an opposite point of view). But at the same time the certainty principle allows to give rigorous formulations for wider class of problems (it allows to rigorously interpret and ground the analogous inequalities for the pairs of quantities like time - energy, angle - angular momentum etc.), Comment: One paper, in Russian and in English, no figures

The electron neutron scattering process determines the electrical and thermal conductivity of neutron stars plasma. As we approach the resonance section of the growth is much higher than the growth predicted by the perturbation theory. The stated mathematical formalism is borrowed from the theory of light which is scattered on the excited system, Comment: in Russian

The work is dedicated to the challenge: control and manipulation of the quantum state of a remote physical object. The goal is approached via the entanglement channel between the object and a local physical system - actuator - subjected to a quantum non-demolishing impact. Physical systems under exploration are light beams., Comment: PhD thesis. in Russian. Research chief: A.I. Lvovsky. 110 pages, 48 figures

This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. It contains an outline of a general heuristic scheme for building univariate special functions through self-similar reductions of spectral transformation chains, which allowed construction of the differential-difference q-Painleve equations, as well as of the most general known set of elliptic biorthogonal functions comprising all classical orthogonal polynomials and biorthogonal rational functions. One of the key results of the thesis consists in the discovery of genuinely transcendental elliptic hypergeometric functions determined by the elliptic hypergeometric integrals. The whole theory of such integrals can be built from the univariate elliptic beta integral -- the most complicated known definite integral with exact evaluation, which generalizes the ordinary binomial theorem and its q-extension, Euler's beta integral, the measure for Askey-Wilson polynomials, and many other previously established results on ordinary and q-hypergeometric functions., Comment: 218 pp, Habilitation Thesis, in Russian, JINR, Dubna, 2004

General key problems in relation to the application of self-organizing structures theory to the analysis of current cooperative phenomena are discussed. Particularly, we present the new results of the analysis of burning problem to form a qubit network. Recently [Proc. of the Inst. of Physics and Technol. of Russian Acad. of Sci. (Trudy FTIAN), vol. 25 (accepted for publication on Jan. 15, 2015)] we have proposed a new method for quantum computer implementation based on self-organizing structures. Here we consider in detail the essence of new proposals and theoretical argumentation. We have shown that some self-organizing structures, both dissipative ones and thermodynamically equilibrium structures with off-diagonal long-range order, are promising to form a qubit network. This is confirmed by our results of the investigations relative to the following items: an open resonator with periodic boundaries, conditions to create specific distribution of the modes and the effect of the field crystallization, the use of this effect to create an proficient and coherentized system of neutral atoms traps and to achieve their precise localization; the equilibrium systems which can go, at temperatures below critical ones, into the special state with off-diagonal long-range order (i.e. to the state with macroscopic quantum coherence), phase transition (photons condensation) in the two-level atoms interacting with radiation mode in the resonator, superradiant phase transition, the order-disorder type ferroelectrics in the cavity electromagnetic field, the advantage of the systems capable to self-organization, including the suppression of the temperature influence as the decoherence factor acting on the work states of the qubits (atoms, ferroelectric molecules), and also the coherent character of the direct dipole-dipole interaction between qubits (in the case of ferroelectrics), and other items., Comment: 39 pages, in Russian

New exact solution class of Born -- Infeld type nonlinear scalar field model is obtained. The variational principle of this model has a specific form which is characteristic for extremal four-dimensional hypersurface or hyper-film in five-dimensional space-time. Obtained solutions are singular solitons propagating with speed of light and having energy, momentum, and angular momentum which can be calculated for explicit conditions. Such solitons will be called the lightlike ones. The soliton singularity has a form of moving two-dimensional surface or shell. The lightlike soliton can have a set of tubelike singular shells with the appropriate cavities. A twisted lightlike soliton is considered. It is notable that its energy is proportional to its angular momentum in high-frequency approximation. A case with one tubelike cavity is considered. In this case the soliton shell is diffeomorphic to a cylindrical surface with threads by multifilar helix. The shell transverse size of the appropriate finite energy soliton can be converging to zero at infinity. The ideal gas of such lightlike solitons with minimal twist parameter is considered in a finite volume. Explicit conditions provide that the angular momentum of each soliton in the volume equals Planck constant. The equilibrium energy spectral density for the solitons is obtained. It has the form of Planck distribution in some approximation. A beam of the twisted lightlike solitons is considered. The representation of arbitrary polarization for the beam with the twisted lightlike solitons is discussed. It is shown that the effect of mechanical angular momentum transfer to absorbent by the circularly polarized beam can be provided. This effect is well known for photon beam. Thus the soliton solution which have determinate likeness with photon is obtained in particular., Comment: 35 pages, 17 figures, final journal version, some clarifications and references added, the source file contains also the text in Russian (see comments in it)

We consider the processes of polarization bremsstrahlung in collisions of fast ions with linear chains consisting of isolated atoms. We obtained intensities and angular distributions of radiation spectra for arbitrary number of atoms in the chain. It appeared that interference in the photon radiation amplitudes lead to prominent variation of spectral angular distributions of polarization bremsstrahlung as compared to these distribuitions in collisions with an isolated atom. The mean loss of energy due to radiation or the so-called rediative friction is estimated. The results obtained permit standard generalization to the case of polarization bremsstrahlung in fast ion chanelling above surfaces an and in solid body., Comment: 7 pages, in Russian

A review of some recently developed methods of calculating multiple differential cross-sections of photoionization and electron impactionization of atoms and molecules having two active electrons is presented. The methods imply original approaches to calculating three-particle Coulomb wave functions. The external complex scaling method and the formalism of the Schroedinger equation with a source in the right-hand side are considered. Efficiency of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. An original numerical method elaborated by the authors for solving the 6D Schroedinger equation for an atom with two active electrons, based on the Chang-Fano transformation and the discrete variable representation, is formulated. Basing on numerical simulations, the threshold behavior of angular distributions of two-electron photoionization of the negative hydrogen ion and helium atom, and multiple differential cross-sections of electron impact ionization of hydrogen and nitrogen molecules are analyzed. It is demonstrated that the Wannier law for the angular distribution of double ionisation is not correct even at very small energies., Comment: In Russian, 74 pages, 14 figures. Submitted to "Physics of Elementary Particles and Atomic Nuclei"

We construct a solution of an analog of the Schr\"{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of variables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schr\"{o}dinger equation corresponding to the Hamiltonian $ H_{II} (z, t, q_1, q_2, p_1, p_2) $ of Hamiltonian system with respect to $t$ which is compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy., Comment: 13 pages, in Russian. There is an English translation of the text containing misprints in the formula (5), (19) and in the first equation of the concluding observations

Here the probability density of relativistic particles coordinates, satisfying the formal conditions of the quantum mechanics and the special relativity, is determined (under textbooks view, such density does not exist). It is specified for the boson, neutral and charged, scalar and vector, including the photon, the electron. Uniqueness of this design has been proved. This is a relativistic invariant, describing the probability of the particle observation in points of space-time. The time is deprived of his role of a dynamic parameter of distribution and formally equalized with other coordinates. Also the meaning of the energy-momentum distribution is clarified. Conditions of such distributions observation are described. As applied to the quantum field these structures are transformed into new characteristics of particles distribution in space-time in addition to those for momentum states. Their operators for free fields are obtained. Under such model the Heisenberg's uncertainty relations are grounded for the relativistic particle; both particles and antiparticles turn out to be different states of a single particle; vacuum state of boson quantum field has zero energy., Comment: 21 pages, in Russian

The conditions for observation of the particle coordinates, required by logic of the Special Relativity and filtering the quantum field effects, are described. A general relation between the corresponding density of probability and the wave function is found. It is a relativistic invariant describing probability of the particle emergences in space -time. This density is concretized for bosons, both scalar and vector (including photon), charged and neutral, also electron. The Heisenberg's uncertainty relations have been approved in regards to relativistic particle. As applied to the quantum field, this new construction is transformed to new characteristic of the particles distribution in space-time, which complete distribution throughout impulses. The operators of these distributions and the invariant relativistic description for free quantum fields have been obtained. These new properties of the particles and fields are proposed for experimental investigations., Comment: 14 pages, in Russian; English translation (14 pages) have been added

Mathematical theory of the quantum systems control is based on some ideas of the optimal control theory. These ideas are developed here as applied to these systems. The results obtained meet the deficiencies in the basis and algorithms of the control synthesis and expand the application of these methods., Comment: 12 pages, in Russian, reference No.15 was corrected

Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed. The main blocks of quantum logic, schemes of implementation of quantum calculations, as well as some known today effective quantum algorithms, called to realize advantages of quantum calculations upon classical, are concerned. Among them special place is taken by Shor's algorithm of number factorization, Grover's algorithm of unsorted database search and, finally, the most promising in application methods of quantum phenomena simulation, particularly quantum chaos. The most perspective methods of experimental realization of quantum computer, namely nuclear-magnetic resonance and trapped ions realizations, are discussed. Phenomena of decoherence, its influence on quantum computer stability and methods of quantum error correction are described., Comment: 6 pages, presented at "Quantum Electrodynamics and Statistical Physics", 2nd Int. Conf., Sept. 19-23, 2006 Kharkov, Ukraine. For extended 28pp. (10pt,wide) version in Russian "Topological Methods In Quantum Computations" (J.KNU,2007) see http://www.math.rutgers.edu/~duplij/publications/Duplij-Shapoval_TOPOLOGICAL-QUANTUM-COMPUTERS.pdf

The construction of oscillator-like systems connected with the given set of orthogonal polynomials and coherent states for such systems developed by authors is extended to the case of the systems with finite-dimensional state space. As example we consider the generalized oscillator connected with Krawtchouk polynomials., Comment: English translation of the article published in Russian

We show that a number of apparently separate extravagant Majorana's actions, including those connected with his disappearance are united by the invariant of behavior. It is based on "carrying into life" the principles of quantum theory. We argue that the underlying motive force in the story of his disappearance consisted in existential intention to overcome the fixed frame of personal identity and dichotomy "life - death" and mimic breakthrough to plurality of worlds., Comment: V.1 in Russian (To appear in Priroda), 16 pages. V. 2 in English, 17 pages

We investigate some dynamical parameters of $\lambda$ - solitons which arises in the family of implicit difference schemes for the diffusion equation with fractional difference operator $ \frac{{\partial ^2}}{{\partial x^2}}$ and imaginary diffusion coefficient. We suppose that such schemes may correspond to fractional diffusion equation with imaginary diffusion coefficient., Comment: 3 pp. in Russian

New status in quantum mechanics is connected with recent achievements in the inverse problem. With its help instead of about ten exactly solvable models which serve as a basis of the contemporary education there are infinite (!) number, even complete sets of such models. So, the whole quantum mechanics is embraced by them. They correspond to all possible variations of spectral parameters which determine all properties of quantum systems. There appears a possibility to change at wish quantum objects by variation of these parameters as control levers and examine quantum systems in different thinkable situations. As a result, we acquire a vision of the intrinsic logic of behavior of any thinkable system, including real ones. The regularities revealed by computer visualization of these models were reformulated into unexpectedly simple universal rules of arbitrary transformations and what is more, their elementary constituents were discovered (new breakthrough). Of these elementary "bricks" it is possible in principle to construct objects with any given properties. This book of inverse problem quantum pictures is utmost intelligible and recommended to any physicists, chemists, mathematicians, biologists from students to professors who are interested in laws of the microworld., Comment: This is the beginning of the new book which is in press now (in Russian, see: http://thsun1.jinr.ru/~zakharev/) First sections have been translated into English (presented here) and will be replenished further on. We have corrected the text (there were some misprints of minor nature)

The monopole systems with hidden symmetry of the two-dimensional Coulomb problem are considered. One of them, the "charge-charged magnetic vortex" with a half-spin, is constructed by reducing the quantum circular oscillator with respect to the action of the parity operator. The other two systems are constructed by reduction from the two-dimensional complex space. The first system is a particle on the sphere in the presence of the exterior constant magnetic field (generated by Dirac's monopole located in its center). This system is dual to the massless (3+1)-dimensional particle with fixed energy. The second system represents the particle on the pseudosphere in the presence of exterior magnetic field and is dual to the massive relativistic anyon., Comment: 10 pages, to be published in Russian J. Nucl. Phys

Using algebraic tools of supersymmetric quantum mechanics we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the help of the raising and lowering operators of these harmonic oscillators and the SUSY operators we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators together with the Hamilton operator form a non-linear algebra which is of quadratic and cubic type for the SUSY partners of the linear and radial harmonic oscillator, respectively., Comment: LaTeX, 11 pages, 3 figures, figures added, minor misprints corrected. to appear in Russian Journal of Nuclear Physics (Yadernaya Fizika)

The marginal distribution of squeezed and rotated quadrature for two types of nonclassical states of trapped ion -- for squeezed and correlated states and for squeezed even and odd coherent states (squeezed Schr\"odinger cat states) is studied. The obtained marginal distribution for the two types of states is shown to satisfy classical dynamical equation equivalent to standard quantum evolution equation for density matrix (wave function) derived in symplectic tomography scheme., Comment: 11 pages,LATEX,contribution to conference Interferometry 2, to appear in Russian Laser Research,1996