Russian Journal of Mathematical Physics最新文献

Abstract

The Pauli–Jordan–Dirac anticommutator mean-square formula is presented.

DOI 10.1134/S1061920823040088

Abstract

In (L_2(mathbb{R}^d;mathbb{C}^n)), we consider a matrix elliptic second order differential operator (B_varepsilon >0). Coefficients of the operator (B_varepsilon) are periodic with respect to some lattice in (mathbb{R}^d) and depend on (mathbf{x}/varepsilon). We study the quantitative homogenization for the solutions of the hyperbolic system (partial _t^2mathbf{u}_varepsilon =-B_varepsilonmathbf{u}_varepsilon). In operator terms, we are interested in approximations of the operators (cos (tB_varepsilon ^{1/2})) and (B_varepsilon ^{-1/2}sin (tB_varepsilon ^{1/2})) in suitable operator norms. Approximations for the resolvent (B_varepsilon ^{-1}) have been already obtained by T.A. Suslina. So, we rewrite hyperbolic equation as a system for the vector with components (mathbf{u}_varepsilon ) and (partial _tmathbf{u}_varepsilon), and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.

DOI 10.1134/S106192082304012X

Abstract

We study a system of two integro-differential equations that arises as the result of linearization of Boltzmann–Maxwell’s kinetic equations, where the collision integral is chosen in the Bhatnagar–Gross–Krook approximation, and the unperturbed state of the plasma is characterized by the Fermi–Dirac distribution. The unknown functions are the linear parts of the perturbations of the distribution function of the charged particles and the electric field strength in plasma. In the paper, an analytical representation for the general solution of this system is found. When deriving this representation, some new results were applied to Fourier transforms of distributions (generalized functions).

DOI 10.1134/S1061920823040039

We study an initial boundary value problem of axially symmetric one-dimensional unsteady shear in the viscoplastic space (a Bingham solid) initiated by a rectilinear vortex thread located along the symmetry axis. The force intensity of the thread is represented by a given monotone piecewise continuous function of time. The density and the dynamical viscosity of the medium are constant, and the yield point is a given piecewise continuous function of radius. We find similar and quasisimilar expressions for the tangent stress and for the rotating component of the velocity both in viscoplastic shear domains and in rigid zones. We show that the vortex thread with time-bounded force intensity may generate a viscoplastic shear only inside a cylinder of certain radius. If the thread intensity growth linearly with time, then the radius of the shear domain grows proportionally to (sqrt t).

We study the operator acting in (L_2(mathbb{R})) by the formula (( mathcal{A} psi)(x)=psi(x+omega)+psi(x-omega)+ lambda e^{-2pi i x} psi(x)), where (xinmathbb R) is a variable, and (lambda>0) and (omegain(0,1)) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate ( mathcal{A} ) using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on ( mathbb{R} ). Within this approach, the analysis of ( mathcal{A} ) turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.