Том 198, № 1 (2019)

To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal field. We propose a type of wave function asymptotically satisfying the operator of initial conditions and still admitting analytic integration of the nonstationary Schrödinger equation. The solution demonstrates that for certain perturbation phases determined by the forcing frequency and the initial indeterminacy of the coordinate, the wave-packet center temporarily stabilizes near the potential maximum for approximately two “natural periods” of the oscillator and then moves to infinity with bifurcations in the drift direction. The effect is not masked by packet spreading, because the packet undergoes anomalous narrowing (collapse) to a size of the order of the characteristic length on the above time interval and its unbounded spreading begins only after this.

Splitting the algebra Psd of pseudodifferential operators into the Lie subalgebra of all differential operators without a constant term and the Lie subalgebra of all integral operators leads to an integrable hierarchy called the strict KP hierarchy. We consider two Psd modules, a linearization of the strict KP hierarchy and its dual, which play an essential role in constructing solutions geometrically. We characterize special vectors, called wave functions, in these modules; these vectors lead to solutions. We describe a relation between the KP hierarchy and its strict version and present an infinite-dimensional manifold from which these special vectors can be obtained. We show how a solution of the strict KP hierarchy can be constructed for any subspace W in the Segal–Wilson Grassmannian of a Hilbert space and any line ℓ in W. Moreover, we describe the dual wave function geometrically and present a group of commuting flows that leave the found solutions invariant.

Based on theWigner unitary representations for the covering Poincaré group ISL(2,ℂ), we construct spin–tensor wave functions of free massive arbitrary-spin particles satisfying the Dirac–Pauli–Fierz equations. We obtain polarization spin–tensors and indicate conditions that fix the density matrices (Behrends–Fronsdal projection operators), which determine the numerators in the propagators of the fields of such particles. Using such conditions extended to the multidimensional case, we construct a generalization of Behrends–Fronsdal projection operators (for any number D >2 of space–time dimensions) corresponding to a symmetric representation of the D-dimensional Poincaré group.

We consider a system of N identical independent Markov processes, each taking values 0 or 1. The system describes the stochastic dynamics of an ensemble of two-level atoms. The atoms are exposed to a photon flux. Under the photon flux action, each atom changes its state with some rates either from its ground state (state 0) to the excited state (state 1) or from the excited state to the ground state (stimulated emission). The atom can also spontaneously change its state from the excited to the ground state. We study rare events where a large cumulative emission occurs during a fixed time interval [0, T]. For this, we apply the large-deviation theory, which allows an asymptotic analysis as N → ∞.

We consider infinite-dimensional unitary principal series representations of the algebra sl_n(ℂ), implemented on the space of functions of n(n−1)/2 complex variables. For such representations, the elements of the Gelfand–Tsetlin basis are defined as the eigenfunctions of a certain system of quantum minors. The parameters of these functions, in contrast to the finite-dimensional case, take a continuous series of values. We obtain explicit formulas that allow constructing these functions recursively in the rank of the algebra n. The main construction elements are operators intertwining equivalent representations and also a group operator of a special type. We demonstrate how the recurrence relations work in the case of small ranks.