Russian Journal of Mathematical Physics最新文献

We consider a difference operator acting in (l^2(mathbb Z)) by the formula (( mathcal{A} psi)_n=psi_{n+1}+psi_{n-1}+lambda e^{-2pi mathrm{i} (theta+omega n)} psi_n), (nin mathbb{Z}), where (omegain(0,1)), (lambda>0), and (thetain [0,1]) are parameters. This operator was introduced by P. Sarnak in 1982. For (omeganotin mathbb Q), the operator ( mathcal{A} ) is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions.

DOI 10.1134/S106192082403004X

In the paper, we consider a linear second order elliptic problem with drift in a domain perforated along the boundary. Setting homogeneous Dirichlet condition on the boundary of the cavities and homogeneous Neumann condition on the outer boundary of the domain, we prove the higher integrability of the gradient of the solution to the problem (the Boyarsky–Meyers estimate).

DOI 10.1134/S1061920824030051

A new practice of applying V.P. Maslov’s theoretical results has been implemented for analyzing fluid flow regimes that arise during their numerical modelling. In this paper, using the example of a Kolmogorov-type flow for two-dimensional motion of a viscous fluid, a rank analysis of the vorticity field and its frequency of occurrence is proposed. A similar analysis has been performed for the problem of forming columnar structures in the spatial case. It has been shown that, for the turbulent, vortex, and laminar fluid motion regimes, the rank distributions exhibit characteristics that can be used to classify the flow types.

DOI 10.1134/S1061920824030075

Left and right-inverse (delta)-suns and left and right (gamma)-suns are studied in asymmetric spaces. Sufficient conditions for the existence of best approximation and solarity of sets are obtained in the uniformly convex asymmetric spaces.

DOI 10.1134/S1061920824030166

The paper deals with the formal short-wavelength asymptotic solutions describing the acoustic eigenoscillations in a vessel having a hard bottom, filled in by an acoustic medium, and covered by a thin elastic membrane. The solutions are localized in the medium near the line of the rigid contact of the membrane covering the vessel with the edge of the vessel. The coefficients in the asymptotic expansion of the solutions satisfy a recurrent sequence of solvable problems, whereas the frequencies, for which such nontrivial formal solutions exist, obey an asymptotic ‘quantization-type condition.

DOI 10.1134/S1061920824030105

In this paper, we introduce the generalized product Hausdorff operator and study the boundedness of this operator on product two-weighted Morrey, Morrey–Herz spaces. As consequences, we obtain some results about the bounds of product Hausdorff operator associated with the Opdam–Cherednik transform and the sharp bounds for the product weighted Hardy–Littlewood average operator and the product Hardy–Cesàro operator on such spaces.

DOI 10.1134/S1061920824030063

The paper deals with the spectral localization in a model problem of singular perturbation theory and the role of the Stokes phenomenon in this context. We study some typical properties of the asymptotic distribution of eigenvalues and, in particular, topologically different types of the spectral configurations in the semiclassical approximation. In this setting the question naturally arises about the corresponding spectral dynamics and the deformation of the actual limit spectral configurations.

DOI 10.1134/S1061920824030026

We study real-valued asymptotic solutions of the discrete Painlevé equation of second type (dPII)

In the case of (n/nu = O(1)), and as (ntoinfty), the asymptotics is nonuniform. Near the point (n= 2nu), an inner transition layer occurs, which matches regular asymptotics to the left and to the right of this point. The matching procedure involves classical Painlevé II transcendents. The asymptotics are applied to discrete gap probabilities and random matrix theory.

DOI 10.1134/S1061920824030130

Using Maslov’s canonical operator in the Cauchy problem for a Dirac equation, we consider the asymptotics of the solution of the Cauchy problem in which the potential depends irregularly on a small parameter.

DOI 10.1134/S1061920824030014