Temperature jump in degenerate quantum gases in the presence of a Bose-Einstein condensate

We construct a kinetic equation modeling the behavior of degenerate quantum Bose gases whose collision rate depends on the momentum of elementary excitations. We consider the case where the phonon component is the decisive factor in the elementary excitations. We analytically solve the half-space boundary value problem of the temperature jump at the boundary of the degenerate Bose gas in the presence of a Bose-Einstein condensate.     

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.In memory of Nikolai Nikolaevich BogolyubovV. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 412–465, December, 1992.     

Smolukhowski problem for degenerate Bose gases

We construct a kinetic equation simulating the behavior of degenerate quantum Bose gases with the collision rate proportional to the molecule velocity. We obtain an analytic solution of the half-space boundary-value Smolukhowski problem of the temperature jump at the interface between the degenerate Bose gas and the condensed phase. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 498–511, June, 2008.     

Partially Classical Limit of the Nelson Model

We consider the Nelson model which describes a quantum system of nonrelativistic identical particles coupled to a possibly massless scalar Bose field through a Yukawa type interaction. We study the limiting behaviour of that model in a situation where the number of Bose excitations becomes infinite while the coupling constant tends to zero. In that limit the appropriately rescaled Bose field converges in a suitable sense to a classical solution of the free wave or Klein-Gordon equation depending on whether the mass of the field is zero or not, the quantum fluctuations around that solution satisfy the wave or Klein-Gordon equation and the evolution of the nonrelativistic particles is governed by a quantum dynamics with an external potential given by the previous classical solution. Communicated by Vincent Rivasseau submitted 20/01/05, accepted 23/01/05     

Scattering of Fermions on a Superfluid Bose Condensate

The influence of a fermionic impurity on the spectrum of overcondensate excitations of a superfluid Bose system is investigated. The scattering of neutrons on the Bose condensate of a superfluid ⁴He and a mixture of the liquid ³He and ⁴He is considered. An analytic expression for the spectrum of overcondensate excitations for these two systems is found. Bibliography: 10 titles.     

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.     

On the dependence of the structure of boundary layers on the boundary conditions in a singularly perturbed boundary-value problem with multiple root of the related degenerate equation

We consider the two-point boundary-value problem for a singularly perturbed secondorder differential equation for the case in which the related degenerate equation has a double root. It is shown that the structure of boundary layers essentially depends on the degree of proximity of the given boundary values of the solution to the root of the degenerate equation; this phenomenon is determined by the multiplicity of the root.     

Quasiaverages and Degenerate Quantum Equilibriums of Magnetic Systems with SU(3) Symmetry of the Exchange Interaction

We consider magnetic systems with the SU(3) symmetry of the exchange interaction. For degenerate equilibriums with broken magnetic and phase symmetries, we formulate classification equations for the order parameter using the concept of residual symmetry. Based on them, we obtain an explicit form of the equilibrium values of the order parameters of a spin nematic and an antiferromagnet in the general form. We clarify the existence conditions for six types of superfluid equilibriums for the order parameter describing the Bose pair condensate. We study inhomogeneous equilibriums and obtain the explicit coordinate dependence of the magnetic order parameters.     

Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

We prove the existence and uniqueness of solution to the nonhomogeneous degenerate elliptic PDE of second order with boundary data in weighted Orlicz‐Slobodetskii space. Our goal is to consider the possibly general assumptions on the involved constraints: the class of weights, the boundary data, and the admitted coefficients. We also provide some estimates on the spectrum of our degenerate elliptic operator.     

Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting

In this note we derive MHD boundary layer equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MHD boundary layer equations. We prove these degenerate boundary layers are stable around a steady solution.     

Thermodynamics of the model of equal spin-spin interactions

We consider the thermodynamics of the model of equal spin-spin interactions. We obtain exact expressions for the correlation functions and heat capacity of finite clusters applicable in the entire range of temperature and external fields. We analyze the obtained thermodynamic characteristics depending on the interaction parameters, the external magnetic field, and the number of particles in the cluster. We find an anomalous behavior of the heat capacity and other thermodynamic quantities due to elementary spin gap excitations occurring in the spectrum. The absence of a long-range order in the system is ensured by the presence of topological excitations (solitons). We study the effect of an anisotropic interaction parameter on the soliton structure.     

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.     

Collective excitations in the superconducting phase of the one-band Hubbard model

The method of functional integration is used to investigate the collective excitations in the superconducting state of the one-band two-dimensional Hubbard model with repulsion. The energy spectrum of the Bose modes is obtained. The existence of a branch of Bogolyubov sound is demonstrated. The phase structure and symmetry of the model are also considered.deceasedSt. Petersburg Branch, V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Metematicheskaya Fizika, Vol. 102, No. 3, pp. 457–462, March, 1995.     

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011) to large volumes.     

Degenerate Nonlinear Boundary Value Problems for Fully Nonlinear Elliptic Equations of Second Order

In this paper, we will consider the degenerate nonlinear boundary value problems for nonlinear elliptic equations. We extend Lieberman & Trudinger's results from nondegenerate oblique derivative boundary values to degenerate boundary values.     

Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains

We establish the existence of the classical solution for the pressure-gradient equation in a non-smooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.     

Diffusion versus absorption in semilinear elliptic equations

We study the limit behaviour of a sequence of singular solutions of a nonlinear elliptic equation with a strongly degenerate absorption term at the boundary of the domain. We give sharp conditions on the rate of degeneracy in terms of the distance to the boundary in order the pointwise singularity not to propagate along the boundary.     

On the Solvability of Double Degenerate Quasilinear Parabolic Equations

We deal with the existence of solutions for double degenerate quasilinear parabolic equations which can degenerate, on a part of the boundary, on a segment in the interior of the domain and in time. This revised version was published online in June 2006 with corrections to the Cover Date.     

Evolutionary weighted p-Laplacian with boundary degeneracy

The paper concerns the well-posedness problem of an evolutionary weighted p-Laplacian with boundary degeneracy. Different from the classical theory for linear equations, it is shown that the degenerate portion of the boundary should be decomposed into two parts: the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics. We formulate reasonably the boundary value condition and establish the existence and uniqueness theorems.