卷 188, 编号 1 (2016)

We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure.

We consider the Hamiltonian H_µ of a system of three identical quantum particles (bosons) moving on a d-dimensional lattice ℤ^d, d = 1, 2, and coupled by an attractive pairwise contact potential µ < 0. We prove that the number of bound states of the corresponding Schrödinger operator H_µ(K), \(K \in \mathbb{T}^d\), is finite and establish the location and structure of its essential spectrum. We show that the bound state decays exponentially at infinity and that the eigenvalue and the corresponding bound state as functions of the quasimomentum \(K \in \mathbb{T}^d\) are regular.

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.

We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field ψ (scalaron). For flat cosmologies (k = 0), we analyze the gauge-independent equation describing the differential χ(α) ≡ ψ (a) of the map of the metric a to the scalaron field ψ, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called a version. In the more customary ψ version, the similar equation for the differential of the inverse map \(\bar \chi (\psi ) \equiv \chi ^{ - 1} (\alpha )\) is solved in an asymptotic approximation for arbitrary potentials v(ψ). In the flat case, \(\bar \chi (\psi )\) and χ⁻¹(α) satisfy first-order differential equations depending only on the logarithmic derivative of the potential, v(ψ)/v(ψ). If an analytic solution for one of the χ functions is known, then we can find all characteristics of the cosmological model. In the α version, the full dynamical system is explicitly integrable for k ≠ 0 with any potential v(α) ≡ v[ψ(α)] replacing v(ψ). Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the α version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for χ. If the conditions for inflation to arise are satisfied, i.e., if v > 0, k = 0, χ² < 6, and χ(α) satisfies a certain boundary condition as α→-∞, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.

We consider the mechanism for generating radiation whose source is formed by surface currents modulated in the variable z - vt in an array of noninteracting parallel nanotubes. The nanotubes are oriented perpendicular to the axis O_z, and the velocity v exceeds the velocity of light in the medium. On the qualitative level, the radiation process under study is analogous to Vavilov–Cherenkov radiation by a system of dipoles. We show that intense SHF and THz radiation can be generated using this method. We estimate the magnitude of millimetric radiation obtainable using an array of nanotubes.