Collective excitations in the ferro-and antiferromagnetic states of the two-dimensional repulsive Hubbard model

Collective excitations in the antiferromagnetic and ferromagnetic states of the two-dimensional repulsive Hubbard model are investigated both in the phase transition vicinity (Ginsburg-Landau region |T−T_c|≪T_c) and in the low-temperature region T≪T_c using the functional integration method. The spectrum of collective excitation consists of one branch at T>T_c, and another branch (Goldstone branch) arises at T<T_c. The Goldstone branch at T≪T_c in the antiferromagnetic state is calculated including the anisotropic correction term to the linear dispersion law. The value of the nonphonon branch at the zero momentum is equal to the double gap of the fermion energy 2Δ. All the spectrum branches in the ferromagnetic state vanish at zero momentum . Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 229–259. Translated by P. A. Sevastianov.     

Magnetic impurity effect on superconductivity in systems with comparable Fermi and Debye energies

We obtain the basic system of equations for a superconductivity theory of a system with a chaotically distributed paramagnetic substitution impurity in which the Migdal theorem is violated (the condition ω_D ≪E _F cannot be assumed). The electron-phonon and impurity diagrams and also the additional diagrams corresponding to intersections of the electron-phonon and electron-impurity lines are taken into account. In the weak electron-phonon coupling limit, we obtain an equation for the superconducting transition temperature T_C that differs from the corresponding equation for the usual superconductors by renormalizations of T_C0 and of the impurity scattering parameter, ρ. These quantities depend essentially on the Migdal parameter ω_D/E _F and on the transferred momentum q_c. We show that the decrease of T_C with the increase of the impurity concentration is slowed, as compared with usual superconductors, to an extent determined by m and q_c. We also evaluate the isotopic coefficient α, whose behavior as a function of the impurity concentration depends on m and q_c. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp 455–474, June, 1999     

Microscopic theory of superconductivity in MgB<Subscript>2</Subscript>-type systems in a magnetic field: A neighborhood of <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis>2</Subscript>

We calculate the upper critical magnetic field H _c2 in the framework of a microscopic superconductivity theory with two energy bands of different dimensions on the Fermi surface with the cavity topology typical of the compound MgB₂ taken into account (an anisotropic system). We assume an external magnetic field parallel to the crystallographic z axis. We obtain analytic formulas in the low-temperature range (T/T_c ≪ 1) and also near the critical temperature ((T-T_c)/T_c ≪ 1). We compare the temperature dependence of H_c2 for a two-band anisotropic system with that of H _c2 ⁰ corresponding to a two-band isotropic system (with Fermi-surface cavities of the same topology). We determine the role of the band-structure anisotropy, the positive curvature of the upper critical field near the critical temperature, and the important role of the ratio v₁/v₂ of the velocities on the Fermi surface in determining H_c2. We also obtain the values of the parameters Δ₁ and Δ₂ along the line of the critical magnetic field. This paper is dedicated to the 90th birthday of Professor D. N. Zubarev __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 113–128, January, 2008.     

Two-band superconductivity theory beyond the Migdal theorem

We propose a superconductivity theory of two-band nonadiabatic systems with strong electron correlations in the linear approximation in nonadiabaticity. Assuming a weak electron-phonon interaction, we obtain analytic expressions for the vertex and “intersecting” functions for each of the two bands. With the diagrams involving intersections of two electron-phonon interaction lines taken into account (which means going beyond the Migdal theorem), we determine mass operators of the Green’s functions and use them to derive the basic equations of the superconductivity theory for two-band systems. We find an analytic expression for the superconducting transition temperature T_c that differs from the expression in the case of the standard two-band systems by an essential renormalization of the relevant quantities that results from the nonadiabaticity effects and strong electron correlations. We study the dependence of T_c and of the isotopic coefficient α on the Migdal parameter m = ω₀/ε_F and show that accounting for the overlap of energy bands on the Fermi surface and for the nonadiabaticity effects at small values of the transferred momentum (q ≪ 2p_F) allows obtaining high values of T_c even for the weak electron-phonon interaction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 111–126, October, 2006. An erratum to this article is available at .     

Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T _c < T _cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T _c or T > T _cr , three weakly periodic Gibbs measures for T = T _c , and one weakly periodic Gibbs measure for T _c < T ≤ T _cr . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 292–302, August, 2008.     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Nonfree actions of countable groups and their characters

The paper studies the region of values of the system {c ₂, c ₃, f(z ₁), f′(z ₁)},where z ₁ is an arbitrary fixed point of the disk |z| < 1; f ∈ T,and the class T consists of all the functions f(z) = z + c ₂ z ² + c ₃z³ + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z ₁) in the subclass of functions f ∈ T with prescribed values c ₂, c ₃, and f(z ₁) is determined. Bibliography: 10 titles.     

Charge density fluctuation of low frequency in a dusty plasma

The charge density fluctuation of low frequency in a dusty plasma, which is derived from the longitudinal dielectric permittivity of the dusty plasma, has been studied by kinetic theory. The results show that theP value, which describes the relative charge density on the dust in the plasma, and the charging frequency of a dust particle Ω _c , which describes the ratio of charge changing of the dust particles, determine the character of the charge density fluctuation of low frequency. For a dusty plasma ofP≪1, when the charging frequency Ω _c , is much smaller than the dusty plasma frequency ω_d, there is a strong charge density fluctuation which is of character of dust acoustic eigenwave. For a dusty plasma ofP≫1, when the frequency Ω _c , is much larger than ω _d there are weaker fluctuations with a wide spectrum. The results have been applied to the ionosphere and the range of radius and density of dust particles is found, where a strong charge density fluctuation of low frequency should exist.     

On simultaneous diophantine approximations. Vectors of given diophantine type

For any monotone functionψ(y)=O(y ^1/s), we prove the existence of a continual family of vectors (α₁...,α_s) admitting infinitely many simultaneous ψ-approximations, but nocψ-approximations with some constantc>0. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 706–716, May, 1997. Translated by S. K. Lando     

Bernstein and Gelfand widths of certain classes of analytic functions. II

The Gelfand widths of the unit ball of H²(ν) (the weighted Hardy space) with respect to the metric of the space L_∞(T_r) are considered (here T_r is the circle of radius r centered at the origin), as well as the Bernstein widths of the unit ball of H^∞ with respect to the metric of the space L₂(T_r, μ). Asymptotic formulas for the widths in question are established for arbitrary measures ν, μ. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 134–140. Translated by S. V. Kislyakov.     

To distortion theorems for typically real functions

The paper studies the region of values of the system {f(z ₁), f(z ₂), c ₂},where z _j , j=1, 2, are arbitrary fixed points of the disk |z|<1; f ∈ T, and the class T consists of all functions f(z) = z + c ₂ z ² + ··· regular in the disk |z| < 1 and satisfying the condition Im f(z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f(z ₁) in the subclass of functions f (z) ∈ T with prescribed values c ₂ and f(z ₂) is determined. Bibliography: 8 titles.     

λφ4model and Higgs mass in standard model calculated by Gaussian effective potential approach with a new regularization-renormalization method

Based on a new regularization-renormalization method, the λφ⁴ model used in standard model (SM) is studied both perturbatively and nonperturbatively by Gaussian effective potential (GEP). The invariant property of two mass scales is stressed and the existence of a (Landau) pole is emphasized. Then after coupling with theSU(2) ×U(1) gauge fields, the Higgs mass in standard model (SM) can be calculated to bem _H≈138 GeV. The critical temperature (T _c ) for restoration of symmetry of Higgs field, the critical energy scale (μ_max, the maximum energy scale under which the lower excitation sector of the GEP is valid) and the maximum energy scale (μ_max, at which the symmetry of the Higgs field is restored) in the SM areT _c ≈476 GeV, μ_c≈0.547 × 10¹⁵ and μ_max≈0.873 × 10¹⁵, respectively. Project supported in part by the National Natural Science Foundation of China.     

On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q^2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ₀ × (0, 2T) of the space-time boundary of the cylinder Q^2T, Γ₀. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 113–136. Translated by T. N. Surkova.     

The closure of the smallest ideal of an ultrafilter semigroup

Let S be a discrete semigroup, let β S be the Stone-Čech compactification of S, and let T be a closed subsemigroup of β S. We characterize ultrafilters from the smallest ideal K(T) of T and from its closure c ℓ  K(T). We show that, for a large class of closed subsemigroups of β S, c ℓ  K(T) is not an ideal of T. This class includes the subsemigroups 0⁺⊂βℝ _d and ℍ _κ ⊂β(⊕ _κ ℤ₂).     

<Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-Singular and <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-singular operators between vector-valued Banach lattices

Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices T _E : E(X) → E(Y) given by T _E (f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c ₀, the operator T is an isomorphism on a subspace isomorphic to c ₀ if and only if so is T _E . An analogous result for invertible operators on subspaces isomorphic to ℓ ₁ is also given.     

Investigation of stress relaxation in polypyromellitimide

A study was made of stress relaxation in polypyromellitimide film at degrees of stretching from 0.5 to 50% over a wide temperature range (below T_c). The applicability of the time-temperature superposition principle is shown at all degrees of stretching. It is shown that the Arrhenius equation for the temperature dependence of the displacement factora _T is valid only at small deformations (up to 1%). The extreme course of the relaxation rate from the level of deformations is attributed to the variations in the polypyromellitimide structure in this deformation region.Scientific-Research Institute of Plastic Compounds, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 45–50, January–February, 1972.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Superconductivity based on a model with three overlapping energy bands and a nonphonon mechanism

A study is made of the thermodynamic properties of a three-band superconductor with an electron (or hole) mechanism of superconductivity. Interband carrier interaction is considered as the main reason for the occurrence of superconducting pairs. Expressions are obtained for the order parameters _n (T) (n=1, 2, 3), the specific heatC _S, and the thermodynamic critical fieldH _c in the region of low temperatures and also the temperatureT _c of the superconducting transition, the jump in the specific heat (C _S–C_N) at the pointT=T _c; and 2 _n (0)/T _c . It is shown that a characteristic feature in the behavior ofT _c, (C _S–C_N), and _n (0) as functions of the carrier concentrationn is the presence of a smeared step. It is also shown that (C _S–C_N)/C _N1.43 and that this ratio depends strongly on the relationship between the densities of the electron states of the considered bands.Institute of Applied Physics, Moldavian Republic Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 483–499, June, 1992.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

The solvability of the initial boundary-value problem for equations of motion of a viscous compressible barotropic liquid in the spacesW 2 l+1,l/2+1 (Q T )

A new method of estimating the solutions of the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded domain Ω⊂ℝ³ is suggested, which makes it possible to investigate the problem for the whole scale of anisotropic spaces W ₂ ^l+2,l/2+1 (Q_T), Q_T=Ω×(0,T), for arbitrary l>1/2. Bibliography: 10 titles. To dear Olga Alexandrovna Ladyzenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 177–186. Translated by V. A. Solonnikov.