Russian Journal of Mathematical Physics最新文献

In [21] and [22], we proved the Kastler–Kalau–Walze type theorem for the (J)-twist (D_{J}) of the Dirac operator on (3)-dimensional, (4)-dimensional, and (6)-dimensional almost product Riemannian spin manifolds with boundary. In this paper, we generalize our previous conclusions and establish the proof of the general Kastler–Kalau–Walze type theorem for the (J)-twist (D_{J}) of the Dirac operator on any even-dimensional almost product Riemannian spin manifold with boundary.

DOI 10.1134/S1061920824601204

We consider the interaction of relativistic particles described by two-dimensional Dirac operators with delta-type singular potentials supported by periodic graphs (Gammasubsetmathbb{R}^{2}). This problem can be regarded as a relativistic analog of the Kronig–Penney model of electron propagation in solid state physics. We associate with this problem an unbounded operator in the Hilbert space (L^{2}(mathbb{R}^{2},mathbb{C}^{2})). The study of spectral properties of these operators is reduced to the study of the Fredholmness of singular integral operators on the graph (Gamma). We obtain necessary and sufficient conditions for the Fredholmness of these operators as ellipticity conditions on the edges, matrix conditions at the vertices, and conditions of invertibility of limit operators which are periodic operators on the graph (Gamma). We apply the Bloch–Floquet theory to the study of invertibility of limit operators.

DOI 10.1134/S1061920825600163

Differential-algebraic descriptions of the orbits of the action of the gauge group for some differential-algebraic sets and functions are constructed. In all cases considered here, it is shown that the orbit is a differential-algebraic set. Applications of the constructed criteria are given.

DOI 10.1134/S1061920825020037

We study the semiclassical Bochner–Schrödinger operator (H_{p}=frac{1}{p^2}Delta^{L^potimes E}+V) on tensor powers (L^p) of a Hermitian line bundle (L) twisted by a Hermitian vector bundle (E) on a Riemannian manifold of bounded geometry. For any function (varphiin C^infty_c(mathbb R)), we consider the bounded linear operator (varphi(H_p)) in (L^2(X,L^potimes E)) defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of (p^{-1}) in the semiclassical limit (pto infty). In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of (varphi(H_p)).

DOI 10.1134/S1061920825600333

We extend the theory of quantum time loops introduced by Greenberger and Svozil [1] from the scalar situation (where paths have just an associated complex amplitude) to the general situation where the time traveling system has multidimensional underlying Hilbert space. The main mathematical tool that emerges is the noncommutative Möbius Transformation and this affords a formalism similar to the modular structure well known to feedback control problems. The self-consistency issues that plague other approaches do not arise here, as we do not consider completely closed time loops. We argue that a sum-over-all-paths approach may be carried out in the scalar case but quickly becomes unwieldy in the general case. It is natural to replace the beam splitters of [1] with more general components having their own quantum structure, in which case the theory starts to resemble the quantum feedback network theory for open quantum optical models and indeed we exploit this to look at more realistic physical models of time loops. We analyze some Grandfather paradoxes in the new setting.

DOI 10.1134/S1061920824601691

It is shown that, in many problems of geometric approximation theory related to min- and max-approximative compactness, it suffices to consider not the entire unit sphere, but rather its only part consisting of acting points (for a given set (M)) — these being the points of the unit sphere such that (M) can be touched by an “analog” of such a point on some homothetic copy of the unit ball. CLUR- and VDS-point for a set are introduced, and their relations to points of min- and max- approximative (norm, weak) compactness are studied. In terms of these points, balayage theorems for problems of min- and max- approximative (norm, weak) compactness of suns and max-suns are obtained.

DOI 10.1134/S1061920825600527

We study boundedly ae-compact sets admitting, for any (varepsilon>0), an (ntau)-continuous (varepsilon)-selection, where (tau) is the topology of convergence in measure. Any such set in (L_p), (1leqslant p<infty), is shown to be a sun. Given a nonempty set, it is shown that the existence of an (ntau)-continuous (varepsilon)-selection for each (varepsilon>0) is equivalent to existence of a norm-norm continuous (varepsilon)-selection for each (varepsilon>0).

We consider a Schrödinger operator on the real line with a super-exponentially decaying and oscillating potential (V(x)=e^{-x^2}big(a-b e^{2 mathrm{i} alpha x}big)), where (a,bin mathbb Csetminus{0}) and (alpha>0) are parameters. Let (k^2) be a spectral parameter. On the complex plane of (k), we find four infinite vertical sequences of resonances of this operator and four finite sequences of resonances located along certain rays in the complex plane. We obtain asymptotic representations for the resonances located far from the origin. The leading terms in the representations are found explicitly, while the error terms are estimated uniformly in (a) and (b). For certain values of the parameters, on the complex plane of (k^2), the vertical sequences might turn into sequences located near the real line, and thus, probably might be interesting for applications in physics.

DOI 10.1134/S1061920825600497

For Volterra integro-differential operators in partial derivatives of the second order, the concept of hyperbolicity with respect to a cone is introduced. It is established that the hyperbolicity with respect to a cone is equivalent to the localization of the support of the fundamental solution of the Volterra integro-differential operator in the conjugate cone. The hyperbolicity with respect to a cone of the integro-differential operator of oscillations of a viscoelastic rod with a fractional-exponential relaxation function is proved.

DOI 10.1134/S106192082502013X

In this paper, we find an upper bound for the first Steklov eigenvalue for a surface of revolution with boundary consisting of two spheres of different radii. Moreover, we prove that, in some cases, this boundary is sharp.

DOI 10.1134/S1061920824601605