The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler _c(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr _c(t) and the pulsation energye(t) it predicts the power lawsr _c(t)t^2/5 andr(t)t ^–6/5.Physics Research Institute of the St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 150–159, July, 1993.     

Superconductivity in two-band systems with a small concentration of carriers

Superconductors with two characteristic features of high-temperature superconducting materials, i.e., a small concentration of charge carriers and overlapping energy bands on the Fermi surface, are considered. At T=0, the order parameters _n and the chemical potential for the Bose condensate of local pairs (<0) is determined in the mean field approximation. The equation for the bound state energy _b is obtained and the relationship _b=2 is established. An application of the path integral method to the two-band model is developed and on this basis, the transition from the Fermi to the Bose pattern of elementary excitations in the presence of a two-particle bound state in the system is demonstrated. The expression for the temperature of Bose condensation T_k is obtained and the contribution of the residual boson interactions is estimated for systems with different dimensions.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 137–151, October, 1996.     

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

We consider the minimization of in a perforated domain of among maps that are incompressible (det ) and invertible, and satisfy a Dirichlet boundary condition u = g on ?Ω. If the volume enclosed by g (?Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B_ε( a _i) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of and the model is suited, therefore, for an asymptotic analysis (v₁,…, v_M denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a ₁,…, a _M, and on the distance from these points to the outer boundary ?Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.     

Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power

We study the stationary Schrödinger equation with the degenerate potential U(x) = x^2r, r Z₊, which describes phase transitions in quantum systems. The symmetry of the problem is revealed and an analytical procedure is developed for finding the eigenvalues of such potentials. The eigenvalues and the lowest energy levels are found numerically for r=2, 3, ..., 18.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 107–123, October, 1996.     

Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: II. The Structure of the Pure Point Spectrum

We study the pure point spectrum of the energy operator H(P ) of a many-particle charged quantum system in a homogeneous magnetic field based on the results in our previous work under fixation of the sum P of the pseudomomentum components of the system. We prove that the discrete spectrum H(P ) of a short-range system is infinite under some conditions (which, for example, hold for a system of two oppositely charged particles) even in the case of a finitely supported potential. For a long-range system of the type of a (+)-ion of an atom (including the ion), the discrete spectrum is infinite.     

Spectral properties of operators of the theory of harmonic potential

We classify the points of the spectrum of the operatorsB andB ^* of the theory of harmonic potential on a smooth closed surfaceS ³. These operators give the direct value onS of the normal derivative of the simple layer potential and the double layer potential. We show that zero can belong to the point spectrum of both operators inL ₂ (S). We prove that the half-interval [–2, 2) is densely filled by spectrum points of the operators for a varying surface; this is a generalization of the classical result of Plemelj. We obtain a series of new spectral properties of the operatorsB andB ^* on ellipsoidal surfaces.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 3–11, January, 1996.This research was partially supported by the Russian Foundation for Basic Research.     

Quasifree States in Some One-Dimensional Quantum Spin Models

We use numerical methods to investigate the SU _q(N) Perk–Schultz spin chain at the special quantum parameter value q=–e ^i/N . We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter q and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers n _i of particles of different types (i=0,1,...,N–1) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of q. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous SU _q(N–1) vertex model that is involved in constructing the system of Bethe-ansatz equations of a matrioshka structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.     

Spectrum of an elliptic equation

It is shown that the spectrum for the first boundary-value problem for a second-order elliptic equation always lies in the half-plane ₀ Re z, where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from ₀, there are no other points of the spectrum on the straight line Re z=₀.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 495–502, April, 1970.The authors are grateful to V. S. Vladimirov for discussing the results of the present paper and for pointing out the proposition in [3], which made it possible to shorten the proof that the leading eigenvalue of the problem (1) is simple.     

Some experimental results on flow in a diverging 2D channel

The effect of the free stream turbulence (FST) on the essential flow characteristics was investigated in the diverging 2D channel. The diverging channel was modelled in the closed type working section by fastening a displacement body above the flat plate that is parallel with the wind-tunnel axis. The angle of the channel expansion 11 degree induced the pressure gradient parameter with the flow velocity U₀ 30 m/s at the start of expansion, x = 0. The height of the channel is 0.15 m at x = 0. FST was either natural 0.3 percent or amplified by turbulence generating grids/screens with turbulence levels up to 5 percent. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the spectrum of the C*-algebra of fourier multipliers in a cone

We describe the spectrum of the C^*-algebra generated by the Fourier multipliers (singular integral operators) on the space L₂ with a polynomial weight in an acute-angled convex cone of the space ⁿ. Bibliography: 12 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 61–82.     

Effective bounds for the number of solutions of certain diophantine equations

It is proved that the number of solutions of the diophantine equation Norm (z ₁, ₁),+...+z _mm) =f (z, ...,z _m), is finite, where ₁, ..., _m are algebraic numbers of a special type, the left side of the equation is the norm with respect to a quadratic field, andf is a low-degree polynomial.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 361–371, September, 1970.     

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.     

Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: I. General Characteristic of the Spectrum

We study the spectrum of Hamiltonians of charged multiparticle systems in a homogeneous magnetic field with a fixed sum P of the pseudomomentum components and without it. We prove that if P is fixed, then the spectrum of Hamiltonians is independent of the value of P , while the spectrum without fixation of P coincides with the spectrum with fixation and differs from the latter only by some additional infinite degeneration (this is a principal difference between problems with a homogeneous magnetic field and problems without any field in which the absence of any fixation of the total angular momentum results in covering the spectrum of the relative motion by a continuous spectrum). We find the continuous spectrum of the Hamiltonians and characterize the spectrum of Hamiltonians of two-cluster mutually noninteracting systems obtained by decomposing the original system in the state with a fixed value of P . The last result is necessary for the study of the purely point spectrum.     

Regularity of a boundary point for singular parabolic equations with measurable coefficients

We investigate the continuity of solutions of quasilinear parabolic equations near the nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for the regularity of a boundary point, which coincides with the Wiener condition for the Laplace p-operator. The model case of the equations considered is the equation with the Laplace p-operator _p for 2n / (n + 1) < p < 2.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 506–516, April, 2004.     

L 2-estimates of solutions of boundary value problems for equations with mixed elliptic-parabolic structure

For the linear equation L u=0 in which L is a product of a 2b₁-parabolic operator and a 2b₂r-elliptic operator (b₁, b₂, and r are integers), we obtain L₂-estimates for solutions of model boundary value problems, namely, for the Cauchy problem and the half-space problem. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 4–27, 1992. Translated by N. A. Karazeeva.     

A kinetic model for polytropic gases with internal energy

We consider a mixture of N ideal, polytropic gases. Each species is described by a distribution function f_i(t, x, v, I) ≥ 0, 1 ≤ i ≤ N, defined on , and its evolution is governed by a Boltzmann-type equation. In order to recover the energy law of polytropic gases, the authors of [4] proposed a kinetic model in the framework of a weighted L¹ space. Another approach has been developed in [3] in the context of polyatomic gases. Following this previous lead, our model provides a L² framework in both variables v and I, to eventually perform a mathematical study of the diffusion asymptotics, as it was done in [2] for a model without energy exchange. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral diffusion tunneling two-level systems

An investigation is made of spectral diffusion due to flip-flop processes in tunneling two-level systems. It is shown that at low frequencies (2k _B T/) of the pulses that saturate the state of the two-level systems and at low temperatures (T<1°K) the mechanism of excitation energy transport through the spectrum of the systems due to flip-flop processes is effective.Institute of Physics, Georgian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 453–458, September, 1993.     

On properties of the discrete and continuous spectrum for the radial dirac equation

To characterize the Dirac radial equation spectrum, we introduce the notion of a quantum defect _k, which is a generalization of this notion for the Schrödinger radial equation. The existence of _k is proved and explicit formulas for calculating _k are found for a broad class of potentials.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 36–49, July, 1996.     

On the schrödinger equation for a flat film with an asymptotically periodic grating

For a potential asymptotically periodic in x ₁ , x ₂ and decreasing at |x ₃ | , the Schrödinger equation is solved and some properties of the solutions are described.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 133–144, January, 1996.