Том 199, № 3 (2019)

We investigate the standard binary Darboux transformation of the supersymmetric Heisenberg model and calculate multisoliton solutions of the quasi determinants of the supersolitons of the Heisenberg magnet model by iterating the Darboux binary transformation.

We propose a model of a spin particle as an analogue of a quantum mechanical top. We show that for this model, we can prove the theorem on the relation between spin and statistics in the framework of nonrelativistic quantum mechanics.

We construct asymptotic eigenfunctions for the two-dimensional Schrödinger operator with a potential in the form of a well that is mirror-symmetric with respect to a line. These functions correspond to librations on this line between two focal points. According to the Maslov complex germ theory, the asymptotic eigenfunctions in the direction transverse to the line with respect to which the well is symmetric have the form of the appropriate Hermite-Gauss mode. We obtain a global Airy-function representation for the asymptotic eigenfunctions in the longitudinal direction.

Our goal is to find and apply simple methods for calculating the scattering cross sections of particles in a screened Coulomb potential in a wider range of wave vector magnitudes than is possible in the Born approximation. Using the technique of Meijer’s G-function, we obtain expressions for the scattering amplitude and the quantum scattering cross section in the limit kd ≫ 1, where d is the screening radius and k is the wavenumber. We compare the analytic results with quantum computations of the elastic scattering process by expanding the solution in a power series of partial waves. We investigate the domain of wave vectors where the Born approximation is no longer applicable but the new approximation works. In this domain, the number of angular momenta that must be taken into account in summing the partial cross sections increases abruptly, and a more precise analytic approximation might therefore noticeably reduce the computational effort. We also show that calculations of the classical integrated scattering cross section must take into account that the finite screening length of an interaction potential causes a cross section to shift off the mass shell. Taking this shift into account allows obtaining a good coincidence of numerical and analytic data for the scattering cross section, including beyond the applicability limits of the Born approximation.

The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.