Volume 191, Nº 2 (2017)

We briefly review the contributions of A. A. Logunov to understanding the problems of general relativity and gravity with special attention to the issue of the possibility of a catastrophic stellar collapse forming a “black hole.”

Based on the relativistic Logunov–Tavkhelidze equation, we obtain the mass spectra and probabilities of radiative decays of heavy quarkonia in the framework of the constituent quark model of hadrons.

The general theory of relativity was first formulated as geometrodynamics, i.e., as a theory where the space–time metric is a dynamic quantity. But it is currently obvious that from the standpoint of particle physics, gravity is simply the theory of a massless spin-2 particle. We show that many important properties of gravity itself and of its possible extensions and modifications can be understood starting from the simple physical requirement that the number of physical degrees of freedom is correct and there are no nonphysical degrees of freedom.

We give a brief review of the most important results on inclusive processes and the relativistic theory of gravity obtained by A. A. Logunov.

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, we define the celestial transformation of Lorentzian vectors, a kind of spinor correspondence. We express a 1-form generating the contact structure in the twistor space (when it is smooth) explicitly as a form taking line-bundle values. We prove a theorem on the projection of this 1-form to the fiberwise normal bundle of a reference frame; its corollary is an equation for the flow of time.

Based on a conservation law, we construct a hodograph transformation for the Wadati–Konno–Ichikawa (WKI) equation, which implies that the WKI equation is equivalent to a modified WKI (mWKI) equation. Applying the Darboux transformation to the mWKI equation, we show that in both the focusing and defocusing cases, the mWKI equation admits an analytic bright soliton solution from the vacuum and the collisions of n solitons are elastic based on the asymptotic analysis. In addition, we find that the mWKI equation still admits the breather and rogue wave solutions, although a modulation instability does not exist for it.

We consider a two-particle Schrödinger difference operator with a periodic potential perturbed by an exponentially decaying interaction potential for particles on a one-dimensional lattice. We obtain rigorous results for the two-particle scattering problem in the case of a small interaction and low velocities. Here, as in other quasi-one-dimensional models, small interactions can significantly affect the scattering pattern. In particular, we find the probability that the velocities of two particles in a periodic medium (e.g., they can be ultracold atoms in a one-dimensional optical lattice) change their signs during a collision. This probability increases as the relative velocity decreases and also as the absolute value of the matrix element between single-particle unperturbed Bloch states increases.

In the Brans–Dicke cosmology framework, we study holographic dark energy models with interaction and with power-law and logarithmic entropy corrections for different cutoffs. We consider conditions on the Brans–Dicke parameter compared with the conditions for the acceleration and phantom phases to show which entropy-corrected models can have acceleration and a phantom phase in the early Universe and at the present. Moreover, we determine which of the considered models are classically stable and which are unstable in early times and at the present.