Russian Journal of Mathematical Physics最新文献

We consider a nonlocal Schrödinger operator on the interval ((0,2pi)) with the periodic boundary conditions and a translation in the free term. The value of the translation is denoted by (a) and is treated as a parameter. We show that the resolvent of such an operator is Hölder continuous in this parameter with the exponent (frac{1}{2},) the spectrum of this operator consists of infinitely many discrete eigenvalues accumulating at infinity, and all eigenvalues are continuous in (ain[0,2pi]) and coincide for (a=0) and (a=2pi.) Our main result is a uniform spectral asymptotics for the operator under consideration. Namely, we show that sufficiently large eigenvalues separate into pairs, each is located in the vicinity of the point (n^2,) where (n) in the index counting the eigenvalues, and we find a four-term asymptotics for these eigenvalues for large (n) with the error term of order (O(n^{-3})), and this term is uniform with respect to (a.) We also discuss nontrivial high-frequency phenomena demonstrated by the uniform spectral asymptotics we have found.

DOI 10.1134/S1061920825600552

We prove that a topological group admitting a family of irreducible unitary representations in Hilbert spaces that separates the elements of the group and whose continuous irreducible representations are finite-dimensional and of bounded degree is a finite extension of a commutative group having sufficiently many continuous characters.

DOI 10.1134/S106192082503015X

We consider a radiation solution (psi) for the Helmholtz equation in an exterior domain in (mathbb{R}^2). We show that (psi) in the exterior domain is uniquely determined by its imaginary part (operatorname{Im}(psi)) on an interval of a line (L) lying in the exterior domain. This result has a holographic prototype in the recent paper by Nair and Novikov (2025, J. Geom. Anal. 35, 4, 123). Some other curves for measurements, instead of the lines (L), are also considered. Applications to the Gelfand–Krein–Levitan inverse problem (from boundary values of the spectral measure in (mathbb{R}^2)) and to passive imaging are also indicated.

DOI 10.1134/S1061920825601077

Commutative positive operator-valued measures (POVMs) are fuzzifications of spectral measures. In quantum mechanics, this corresponds to a connection between commutative unsharp observables (represented by commutative POVMs) and sharp observables (represented by self-adjoint operators); the former being a fuzzification of the latter. We prove that commutative unsharp observables can be defuzzified in order to obtain the sharp observables of which they are the fuzzy versions. We prove this in the case of POVMs defined on a general topological space which we require to be second countable and metrizable, generalizing some previous results on real POVMs. Then, we analyze some of the consequences of this defuzzification procedure. In particular we show that the joint measurability of two commutative (but generally not commuting) POVMs (F_1) and (F_2) corresponds to the existence of two commuting self-adjoint operators (A^+_1) and (A^+_2) in an extended Hilbert space (mathcal{H}^+) whose projections are the sharp versions of (F_1) and (F_2), respectively. In other words, the joint measurability of (F_1) and (F_2) is translated in the commutativity of (A_1^+) and (A_2^+). This is proved for POVMs on a second countable, Hausdorff, locally compact topological space, generalizing similar results obtained in the case of real POVMs.

DOI 10.1134/S1061920825600692

Quantum filtering equations for mixed states were developed in the eighties of the last century. Since then, the problem of constructing a rigorous mathematical theory for these equations in the basic infinite-dimensional settings has been a challenging open mathematical problem. In a previous paper, the author developed the theory of these equations in the case of bounded coupling operators, including a new version that arises as the law of large numbers for interacting particles under continuous observation and thus leading to the theory of quantum mean field games. In this paper, the main body of these results is extended to the basic cases of unbounded coupling operators.

DOI 10.1134/S1061920825600825

We consider the Lavrentiev–Bitsadze equation in partially perforated domain. Under the assumption of stochastic geometry of the domain we derive the homogenized equation and prove the convergence of solutions of an original problem to the solution of the homogenized problem.

DOI 10.1134/S1061920825600813

Equations of the double-deck boundary layer structure are obtained in the problem of a flow of a viscous incompressible fluid around a small irregularity on a curved surface at high Reynolds numbers. It is shown that ,due to the chosen coordinate system, the form of the equations of the double-deck structure coincides with those of the previously studied case of a small irregularity on a flat surface; the difference lies only in the values of the coefficients. This means that the results of flow modelling for the flat case can be qualitatively transferred to the curvilinear case.

DOI 10.1134/S1061920825600461

We consider a class of infinite vector continued fractions of a complex variable such that their coefficients are bounded operators in a Hilbert space. They may be regarded as analogs (in a broad sense) of (J)-fractions used in the theory of Jacobi operators and the classical moment problem. To each of the continued fractions under consideration, there corresponds the band operator generated by certain infinite block matrix containing a finite number of nonzero diagonals, which are composed of the (operator) elements of this continued fraction. Using the inverse spectral theory for these band operators, we establish the main properties of such continued fractions, in particular, their expansion algorithm and the criterion for existence of this expansion. It turns out that the algorithm of reconstruction of a band operator from its spectral data (the moment sequence of its Weyl function) can be regarded as a modified version of a known Jacobi-Perron expansion algorithm, applied to a system of operator-functions holomorphic at infinity in order to get a continued fraction from the class under study. Certain issues of the theory of Hermite-Padé approximants, related to the studied subject, are also considered.

DOI 10.1134/S1061920825030148

We prove that a unitary representation (rho) of an amenable locally compact group (G) such that (|rho(g)-pi(g)|le q<1/2) for all (gin G) and for some continuous unitary representation (pi) of (G) in the same Hilbert space is unitary equivalent to (pi).

DOI 10.1134/S1061920825020153

In this paper we study a wave equation whose velocity has a localized perturbation at some point (x_0). The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable with the scale of the inhomogeneity. In this case, the length of the initial wave is of the order of (varepsilon), and the width of the localized inhomogeneity is of the order of (varepsilon^{1/m},) where (varepsilon) is a small parameter that tends to 0, and (m) is a positive integer greater than 2.

DOI 10.1134/S1061920825600916