Theoretical and Mathematical Physics最新文献

We study the dependence of the beta-function on the running coupling constant in holographic models supported by the Einstein–dilaton–Maxwell action for light and heavy quarks. The dilaton defines the running coupling of the models. Its dependence on boundary conditions leads to the running coupling dependence on them. However, the behavior of the (beta)-function as a function of the running coupling does not depend significantly on the boundary condition. The corresponding (beta)-functions are negative and monotonically decreasing functions, and have jumps at first-order phase transitions for both light and heavy quarks. In addition, we compare our holographic results for the (beta)-function as a function of the running coupling with perturbative results obtained within (2)-loop calculations.

We reformulate the time-independent Schrödinger equation as a Maurer–Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential such that its cohomology becomes the space of solutions with a fixed energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.

An extended (2)-dimensional Toda lattice equation is investigated by means of the Cauchy matrix approach. We introduce a direction parameter in the extension and represented the equation as a coupled system in a (3)-dimensional space. The equation can also be considered as a negative-order member in one direction of the discrete Kadomtsev–Petviashvili equation. By introducing the (tau)-function and an auxiliary direction, the equation can be bilinearized in a (4)-dimensional space with a single (tau)-function.

We introduce the notion of Darboux transformations for the discrete KP hierarchy and its strict version, and present an explicit form of these transformations for the solutions of discrete KP and discrete strict KP hierarchies constructed in our previous work.

We explore the dynamics of the Klein–Gordon oscillator in the presence of a cosmic string in the Som–Raychaudhuri space–time. The exact solutions for the free and oscillator cases are obtained and discussed. These solutions reveal the effects of the cosmic string and space–time geometry on bosonic particles. To illustrate these results, several figures and tables are included.

We show that the Lie algebraic approach with the perturbation method can be used to study the eigenvalues of the Hellmann Hamiltonian. The key element is the Runge–Lenz vector, which appears in problems with radial symmetry. This symmetry implies that the proper lie algebra for these Hamiltonians is (so(4)), which is a sum of two (so(3)) Lie algebras and requires symmetry of the angular momentum vector (vec{L}) and the Runge–Lenz vector (vec{M}), and therefore their cross products as (vec{W}=vec{L}timesvec{M}). Here, Yukawa potential is considered as a perturbation term, which is added to the Coulomb Hamiltonian to produce the Hellmann Hamiltonian. Lie algebraically, the perturbation term adds a magnitude of precession rate (Omega) to all three operators (vec{L}), (vec{M}), and (vec{W}). Topologically, we show that the appearance of this precession has a significant effect on the spectrum and the corresponding Lie algebra of the Hellmann potential. By using Lie algebraic properties of the Runge–Lenz vector and using the Kolmogorov method, we obtain the energy spectrum of this Hamiltonian.

We formulate an inverse scattering transformation for the focusing Hirota equation with asymmetric boundary conditions, which means that the limit values of the solution at spatial infinities have different amplitudes. For the direct problem, we do not use Riemann surfaces, but instead analyze the branching properties of the scattering problem eigenvalues. The Jost eigenfunctions and scattering coefficients are defined as single-valued functions on the complex plane, and their analyticity properties, symmetries, and asymptotics are obtained, which are helpful in constructing the corresponding Riemann–Hilbert problem. On an open contour, the inverse problem is described by a Riemann–Hilbert problem with double poles. Finally, for comparison purposes, we consider the initial value problem with one-sided nonzero boundary conditions and obtain the formulation of the inverse scattering transform by using Riemann surfaces.

Earlier, we have presented an integrable system with a negative time variable number for the Davey–Stewartson hierarchy. Here, we develop this approach to construct an integrable equation with a lower time variable number. In addition, we show that the system reduced with respect to this time yields a new integrable equation in (1+1) dimensions.

In the nonisospectral case, we introduce the associated spectral problem with a perturbation term. We obtain a generalized nonisospectral super AKNS hierarchy and a coupled generalized nonisospectral super AKNS hierarchy associated with generalized Lie superalgebras (sl(2,1)) and (sl(4,1)). Based on a new type of multicomponent Lie superalgebra (sl(2N,1)), a multicomponent generalized nonisospectral super AKNS hierarchy is obtained. By using the supertrace identity, the super bi-Hamiltonian structures of the resulting superintegrable hierarchies are obtained.

We study the spectrum of the Landau Hamiltonian with a periodic electric potential. In the case of a rational magnetic flux, we present examples of nonconstant zero-mean periodic electric potentials ({Vin C^{infty}(mathbb{R}^2;mathbb{R})}) for which the spectrum has an eigenvalue at the second Landau level.