Spectral behavior of quasi periodic potentials

Spectral properties of Schrödinger operators of the typeH =–+V, where is the Laplacian,V a quasiperiodic potential and a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as is increased.Research supported in part by grant NSF MCS 7900813     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

P-polarized nonlinear surface polaritons near the surface of a saturable self-defocusing medium: Threshold case

We have obtained the solution of the stationary problem for P-polarized nonlinear surface polaritons (NLSP) near the interface of two media, one of which is optically linear and the other having an optical nonlinearity with saturation and a dielectric function of the form: =_xx=_zz=₀+a|E|²/(1+b|E|²), where E is the amplitude of the electric field. On this basis the NLSP energy and propagation properties, the polarization structure and the localization depth of the NLSP field as well as the physically allowed region in the parameter space have been investigated for the case in which the parameter ₀ is positive, b is non-negative, and a and ₁ are negative (₁ is the dielectric function of the linear medium); ₀>|₁|.     

Scattering states and bound states in λ℘(φ)2

By analyzing the Bethe-Salpeter equation for even ()₂ models we show that for weak coupling the mass spectrum is discrete and of finite multiplicity below 2m. Moreover on even states of energy less than 4(m–) we show that theS matrix is unitary. Herem is the physical mass and =()0 as 0. Our results rely essentially only on a simple assumption about the analyticity of the Bethe-Salpeter kernel which has been verified for weak coupling. For the interaction ⁴, (/m _o ² 1) we show that there are no even bound states of energy less than 4(m–).Work supported in part by NSF, Grant MPS 74-13252     

Kinetic theory of adsorption and desorption

In this paper we use stochastic methods to discuss adsorption and desorption. The paper derives generalized coefficients of sticking and accomodation depending on surface temperatureT _s and gas temperatureT _g and shows, that for additive Markov processes, these kinetic coefficients are identical. Furthermore, exact solutions of the kinetic equations for certain simple transition probabilitiesP(, ) are found and an approximation method for more complicatedP(, ) is given. The comparison of the theory with experimental results for noble gas-metal systems indicate a quadratic relationship between the first moment of the transition probabilityP(, ) and the well depth of the physisorption system.     

Kinetic limits of the HPP cellular automaton

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings, ^- stirring updatings, <1 being any fixed positive number and a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order of (low-density limit) and varying on distances of the order of ^–1. The limit as 0 of the system evolved for times of the order of ^-1- corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided by) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.     

Partial diagonalization in solving electron-nuclear problems in molecules

The problem of determining the total wave functions and energies of molecular stationary states reduces to solving a Schrödinger equation with a vibrational-rotational Hamiltonian. This is achieved by a unitary transformation of the molecular Hamiltonian H with its successive diagonalization on a nondegenerate electronic state ¦e. It is shown that the molecular wave functions related to the electronic states ¦e are of the form G¦e¦g_(e), and their corresponding energy value is the sum _e + _g ^(e), where _g ^(e) and ¦g^(e) are the eigenvalues and eigenfunctions of the vibrational-rotational Hamiltonian, determined by means of the unitary operator G. It is shown that the total energy and molecular wave functions are uniquely determined, despite the arbitrariness in choosing G. As an example the vibrational-rotational operator and molecular wave functions are given for the simplest choice of the operator G.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 11–16, March, 1975.     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.     

ИЗЛУЧЕНИЕ ЭЛЕКТРИЧЕ СКОГО ДИПОЛЯ НАД ГРАН ИЦЕЙ ДВУХ СРЕД

, , . , - - . - . -, - , . , - ; , , .     

Determination of electrodynamical parameters of dielectric pipe-shaped materials using millimeter wave cavity

Employment ofE _0np modes of cylindrical cavity resonator for measurement of electrodynamical parameters of dielectric pipe-shaped materials in millimeter wavelength band is suggested. Dielectric permittivity and conductivity of a sample can be found by measuring the frequency shift and theQ-factor change of the resonator.Method of mode basis is used for the calculation of sample permittivity and conductivity . Measurements were carried out at 8mm wavelength and related calculations have demonstrated use and possibilities of the method in studying the dielectric properties of samples of cylindrical and other forms.     

Natural boundaries for area-preserving twist maps

We consider KAM invariant curves for generalizations of the standard map of the form (x, y)=(x+y, y+f(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation +. In the complex plane, natural boundaries of different shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as tends to the critical value.     

Accurate calculation of functional determinants

A dependence of the functional determinant of the operator from the family D() satisfying the condition D()=fD()+D()f on the parameter m² of infrared regularization is found in the regularization method using a generalized function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 27–32, June, 1988.     

Correlations and spectra of an intermittent chaos near its onset point

A one-parameter family of piecewise-linear discontinuous maps, which bifurcates from a periodic state of periodm, (m=2, 3,...) to an intermittent chaos, is studied as a new model for the onset of turbulence via intermittency. The onset of chaos of this model is due to the excitation of an infinite number of unstable periodic orbits and hence differs from Pomeau-Manneville's mechanism, which is a collapse of a pair of stable and unstable periodic orbits. The invariant density, the time-correlation function, and the power spectrum are analytically calculated for an infinite sequence of values of the bifurcation parameter which accumulate to the onset point _c from the chaos side - _c > 0. The power spectrum near=0 is found to consist of a large number of Lorentzian lines with two dominant peaks. The highest peak lies around frequency=2/m with the power-law envelope l/¦-(2/m)¦⁴. The second-highest peak lies around o = 0 with the envelope l/¦¦². The width of each line decreases as, and the separation between lines decreases as/lg3^–1. It is also shown that the Liapunov exponent takes the form-/m and the mean lifetime of the periodic state in the intermittent chaos is given bym ^–1(ln ^–1+1).     

Resonant shielding of an inhomogeneous gyrotropic plasma slab

Conclusions The problem of the incidence of a plane TM electromagnetic wave on an isotropic, symmetric, resonance plasma slab (n=2, 4, 6, ...), discussed in [1], was solved by an iteration method in a recent paper [6]. The physical results found there are the same as those of [1]. Zhivulin and Makarov [6] then applied the iteration method to the analogous problem of a gyrotropic resonance plasma slab [7]. The analysis in these papers furnishes a clearer mathematical justification of the results of [1] and the present paper and thus of the method used there. The present method, which satisfies only a physical condition of rigor, is preferable to the mathematically more rigorous methods (in particular, the iteration method) because of its simplicity, its graphic nature, and its clear physical meaning. It also answers many questions which cannot be answered in the more rigorous approach because of the serious difficulties which arise (and which have not yet been overcome).N. G. Denisov has called our attention to the fact that complete shielding was actually found previously by Rytov and Yudkevich [8], who treated the problem of the incidence of a plane TE electromagnetic wave on a slab with a dielectric constant (x)=₁ – A₁/(a – x)² for x<0. (x)=₂ – A₂/(b + x)² for x>0, and ₁ – A₁/(a² = ₂ – A₂/b² in the plane x=0. In the limita0, b0, they found results corresponding to a slab with a dielectric constant having a first-order pole; it is in this case that complete shielding is achieved. This method for obtaining the corresponding results is analogous to the method used in [1] and the present paper. We also note that the distribution of the effective dielectric constant (6) in the immediate vicinity of the pole—where the contribution of the last term. (1–u)x², can be neglected—is the same as the distribution adopted in [8] if we seta-b-0, ₁ = ₂, A₁=A₂=u.Scientific-Research Institute of Radiophysics. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 19, No. 8, pp. 1130–1141, August, 1976.     

Electromagnetic waves in thin-film superconducting lines

Conclusions The distinctive features discussed about the propagation of electromagnetic waves in a structure containing a thin-film superconducting line are associated with the fact that the screening properties of the superconducting layer are determined by the distribution of superconducting currents in it. In the quasistatic case under discussion this distribution depends on the nature of the nonuniformity of the magnetic field along the normal to the layer. At specified angles of incidence of a wave the layer is equivalent to an ideal conductor or a diamagnetic; in the intermediate region the screening effect is far smaller. Due to this a thin-film superconducting line possesses some directionality of the leakage of the nonuniform waves when ₁ > ₂. The wave which is emergent at a specified angle which depends on the parameters of the layered structure (₁, ₂,l, _L) possesses the maximum amplitude.When ₁ < ₂, the structure in question does not have a sharply pronounced directionality of the leakage, and at angles of incidence on a thin layer corresponding to total internal reflection it supports the propagation of waves along a line with as small a film thickness as desired. The unscreening effect of a superconducting layer is exhibited here in the fact that the propagation of waves excited in the line occurs at specified angles of incidence onto the layer mainly along its outer surface.Institute of Applied Physics, Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Vol. 25, No. 5, pp. 527–535, May, 1982.     

On the hole spectral function of the Emery model

We discuss the spectral function of a single O hole generated in a two-dimensional CuO₂-lattice at half-filling. The latter constitutes the most important structural element of high-T _c superconducting materials. The system is described by the so-called extended Hubbard or Emery model. Strong electronic correlations which are incorporated in the model prevent the usual evaluation of Green functions based on Wick's theorem and using diagram techniques. For that reason we apply a new cumulant approach to dynamical correlation functions introduced recently. As a result we find that the local one-O hole excitation spectrum has two structured absorption regions around the bare O energy _p and around _p + due to charge fluctuations of Cu holes. Here is the bare charge transfer gap. The width of the absorption regime around _p is of the order of several timest _pd ² /, wheret _pd is the hopping integral between Cu and O holes.     

Changes in the properties of dielectric material T-8000 produced by irradiation with electrons and protons

In studying the dielectric properties of material T-8000, a difference was discovered in the effects of electron and proton irradiation on dielectric permeability () and dielectric loss angle tangent (tan ). It is assumed that upon irradiation with protons having higher energy (4.25 MeV) the processes of titanium, zirconate, and barium ion displacement occur with greater probability than upon irradiation with electrons whose energy does not exceed 1.2 MeV. This causes the differences in the changes in tan and of the T-8000 material.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 46–49, March, 1974.     

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

Self-consistent perturbation theory for dynamics of valence fluctuations

Dynamical properties of valence-fluctuating systems are studied at absolute zero of temperature. The self-consistent perturbation theory developed for rare-earth impurity systems is used with some refinement. The theory takes account of the orthogonality catastrophe caused by hybridization of 4f and conduction electrons. Extensive numerical results are reported for the 4f-electron density of states _4f () and the dynamical magnetic susceptibility (). The results cover both the intermediate-valence and nearly integral-valence regimes of a model Ce impurity system. The present theory gives reasonable overall feature of dynamics including a sharp peak in _4f (0) in the Kondo regime. However, the low-frequency limit of dynamical quantities is not consistent with the Fermi-liquid property. It is shown that interpolation of the present results and those by the Fermi-liquid theory is possible. Hence accurate dynamical information is obtained over a wide excitation-frequency range.     

Stationary states of an electron in the field of a hole in an ionic crystal

An approximate solution is given of the Schrödinger equation for S-states of an electron in the field of a hole, when the potential energy of the electron has the form — e²/r [1+exp (–qr].

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s- , –e²/r [1+exp (–qr].