Том 195, № 1 (2018)

We consider differential–difference equations defining continuous symmetries for discrete equations on a triangular lattice. We show that a certain combination of continuous flows can be represented as a secondorder scalar evolution chain. We illustrate the general construction with a set of examples including an analogue of the elliptic Yamilov chain.

We consider the Schrödinger equation with an additional quadratic potential on the entire axis and use the transformation operator method to study the direct and inverse problems of the scattering theory. We obtain the main integral equations of the inverse problem and prove that the basic equations are uniquely solvable.

Let ϕ be a trace on the unital C*-algebra A and M_ϕ be the ideal of the definition of the trace ϕ. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P − Q ∈ M_ϕ, then ϕ((P − Q)²ⁿ⁺¹) = ϕ(P − Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M_ϕ with ϕ(U-A) ∈ R. Then I-A, I−U ∈ M_ϕ and ϕ(I−U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U − V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.

We obtain an expression for the neutron scattering cross section in the case of an arbitrary interaction of the neutron with the crystal. We give a concise, simple derivation of the Debye–Waller factor as a function of the scattering vector and the temperature. For ferromagnetic metals above the Curie temperature, we estimate the Debye–Waller factor in the range of scattering vectors characteristic of polarized magnetic neutron scattering experiments. In the example of iron, we compare the results of harmonic and anharmonic approximations.

We provide a complete classification of static plane symmetric space–times according to conformal Ricci collineations (CRCs) and conformal matter collineations (CMCs) in both the degenerate and nondegenerate cases. In the case of a nondegenerate Ricci tensor, we find a general form of the vector field generating CRCs in terms of unknown functions of t and x subject to some integrability conditions. We then solve the integrability conditions in different cases depending upon the nature of the Ricci tensor and conclude that the static plane symmetric space–times have a 7-, 10- or 15-dimensional Lie algebra of CRCs. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We use a similar procedure to study CMCs in the case of a degenerate or nondegenerate matter tensor. We obtain the exact form of some static plane symmetric space–time metrics that admit nontrivial CRCs and CMCs. Finally, we present some physical applications of our obtained results by considering a perfect fluid as a source of the energy–momentum tensor.

We study thermal quantum correlations (quantum discord and super quantum discord) in a two-spin model in an external magnetic field and obtain relations between them and entanglement. We study their dependence on the magnetic field, the strength of the spin squeezing, and the temperature in detail. One interesting result is that when the entanglement suddenly disappears, quantum correlations still survive. We study thermal quantum teleportation in the framework of this model. The main goal is investigating the possibility of increasing the thermal quantum correlations of a teleported state in the presence of a magnetic field, strength of the spin squeezing, and temperature. We note that teleportation of quantum discord and super quantum discord can be realized over a larger temperature range than teleportation of entanglement. Our results show that quantum discord and super quantum discord can be a suitable measure for controlling quantum teleportation with fidelity. Moreover, the presence of entangled states is unnecessary for the exchange of quantum information.