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Том 199, № 2 (2019)
- Год: 2019
- Статей: 11
- URL: https://journals.rcsi.science/0040-5779/issue/view/10494
Article
Factorization of Darboux—Laplace Transformations for Discrete Hyperbolic Operators
Аннотация
We classify elementary Darboux—Laplace transformations for semidiscrete and discrete second-order hyperbolic operators. We prove that there are two types of elementary Darboux—Laplace transformations in the (semi) discrete case as in the continuous case: Darboux transformations constructed from a particular element in the kernel of the initial hyperbolic operator and classical Laplace transformations that are defined by the operator itself and are independent of the choice of an element in the kernel. We prove that on the level of equivalence classes in the discrete case, any Darboux—Laplace transformation is a composition of elementary transformations.
621-636
637-651
Quasi-Stäckel Hamiltonians and Electron Dynamics in an External Field in the Two-Dimensional Case
Аннотация
In the two-dimensional case, we construct nondegenerate Hamiltonians that describe electron motion in an electromagnetic field and have additional integrals of motion quadratic in momentum. We completely classify the quasi-Stäckel Hamiltonians related to these systems in the cases where the leading approximation in momenta of the additional integral depends quadratically on the coordinates. We consider reductions of such systems that are symmetric under rotation about the z axis.
652-658
Superintegrable Systems with Algebraic and Rational Integrals of Motion
Аннотация
We consider superintegrable deformations of the Kepler problem and the harmonic oscillator on the plane and also superintegrable metrics on a two-dimensional sphere, for which the additional integral of motion is either an algebraic or a rational function of momenta. According to Euler, these integrals of motion take the simplest form in terms of affine coordinates of elliptic curve divisors.
659-674
Algebro-Geometric Integration of the Modified Belov—Chaltikian Lattice Hierarchy
Аннотация
Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov—Chaltikian lattice hierarchy associated with a discrete 3×3 matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a tri gonal curve Km−2 of arithmetic genus m−2. We study the asymptotic properties of the Baker—Akhiezer function and the algebraic function carrying the data of the divisor near \(P_{\infty_{1}}\), \(P_{\infty_{2}}\), \(P_{\infty_{3}}\), and P0 on Km−2. Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker—Akhiezer function, and, in particular, solutions of the entire modified Belov—Chaltikian lattice hierarchy.
675-694
Bosonic Symmetries of the Extended Fermionic (2N, 2M)-Toda Hierarchy
Аннотация
We construct the additional symmetries of the fermionic (2N, 2M)-Toda hierarchy based on a generalization of the N=(1∣1) supersymmetric two-dimensional Toda lattice hierarchy. These additional flows constitute a w∞×w∞ Lie algebra. As a bosonic reduction of the N =(1∣1) supersymmetric two-dimensional Toda lattice hierarchy and the fermionic (2N, 2M)-Toda hierarchy, we define a new extended fermionic (2N, 2M)-Toda hierarchy that admits a bosonic Block-type superconformal structure.
695-708
Dressing Method for the Multicomponent Short-Pulse Equation
Аннотация
We use the Zakharov—Shabat dressing method to solve the multicomponent short-pulse equation. We obtain dressed solutions of this equation using a Riemann—Hilbert problem. The dressed solutions of the Lax pair and the multicomponent short-pulse equation are expressed in terms of Hermitian projectors. We show that the dressed solutions are related to quasideterminant solutions, and we write K-soliton solutions in terms of quasideterminants. In explicit form, we obtain one- and two-soliton solutions.
709-718
Quantum Problem of Polaron Localization and Justification of the Su—Schrieffer—Heeger Approximation
Аннотация
We consider a polaron localized on a trap in a one-dimensional lattice with a harmonic potential of nearest-neighbor interaction. We study the lattice oscillations in the framework of quantum mechanics and regard the polaron energy as a functional of the wave function. We take the electron—phonon interaction into account in the framework of the linear Su—Schrieffer—Heeger approximation. We show that the results do not differ from the adiabatic consideration if the lattice oscillations are described classically. For the polaron ground state, we obtain approximate analytic expressions that agree well with the results of numerical simulation.
719-725
Translation Invariance of the Periodic Gibbs Measures for the Potts Model on the Cayley Tree
Аннотация
We study the Potts model with a zero external field on the Cayley tree. For the antiferromagnetic Potts model with q states on a second-order Cayley tree and for the ferromagnetic Potts model with q states on a kth-order Cayley tree, we show that all periodic Gibbs measures are translation-invariant for all parameter values.
726-735
Nonchiral Bosonization of Strongly Inhomogeneous Luttinger Liquids
Аннотация
Nonchiral bosonization (NCBT) is a nontrivial modification of the standard Fermi—Bose correspondence in one spatial dimension done to facilitate studying strongly inhomogeneous Luttinger liquids where the properties of free fermions plus the source of inhomogeneities are reproduced exactly. We introduce the NCBT formalism and discuss limit case checks, fermion commutation rules, point-splitting constraints, etc. We expand the Green’s functions obtained from NCBT in powers of the fermion—fermion interaction strength (only short-range forward scattering) and compare them with the corresponding terms obtained using standard fermionic perturbation theory. Finally, we substitute the Green’s functions obtained from NCBT in the Schwinger—Dyson equation, which is the equation of motion of the Green’s functions and serves as a nonperturbative confirmation of the method. We briefly discuss some other analytic approaches such as functional bosonization and numerical techniques like the density-matrix renormalization group, which can be used to obtain the correlation functions in one dimension.
736-760
761-770