THE FINITENESS OF THE NUMBER OF EIGENVALUES OF THE FOUR-PARTICLE SCHRÖDINGER OPERATOR WITH THREE-PARTICLE CONTACT INTERACTION

In this paper, we consider the four-particle Schr\"{o}dinger operator corresponding to the Hamiltonian of a system of four arbitrary quantum particles via a three-particle contact interaction potential on a three-dimensional lattice. The finiteness of the number of eigenvalues of the Schr\"{o}dinger operator lying to the left of the essential spectrum for zero value of the total quasi-momentum is proved.

Quasi-periodic Wannier–Stark ladders from driven atomic Bloch oscillations

Periodic Wannier–Stark ladder structures of the energy resonances associated with Bloch oscillations can be readily modified into quasi-periodic ones that exhibit peculiar self-similar effects. A compact theoretical description of the dynamics of driven Bloch oscillations is developed here within the quasi-momentum representation. We identify a rather viable scheme based on ultracold atomic wavepackets subject to gravity in a driven optical lattice potential where a self-similar scaling could be observed. Its feasibility in terms of realistic experimental parameters is also discussed.

ON THE SPECTRUM OF A HOLOMORPHIC OPERATOR-VALUED FUNCTION WITH APPLICATIONS TO ABSORPTIVE PHOTONIC CRYSTALS

We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.

SPECTRA AND BOUND STATES OF THE ENERGY OPERATOR OF TWO-MAGNON SYSTEM IN A NON-HEISENBERG FERROMAGNET WITH SPIN S = 3/2 AND NEAREST-NEIGHBOR INTERACTIONS

We consider a two-magnon systems in an ν-dimensional isotropic non-Heisenberg ferromagnet with spin value S = 3/2 and nearest-neighbor interactions. Spectrum and bound states (BS) of the system for all values of full quasi-momentum Λ, and for arbitrary value of lattice dimensionality ν, and for all values of Hamiltonian parameters are investigated. We show that (i) for arbitrary ν ≥ 2 and for full quasi-momentum in the form Λ = (Λ1; Λ2; … ;Λν) = (Λ0;Λ0; …; Λ0) the change of energy spectrum of the system is similar to that observed in the case of ν = 1. In this case the operator [Formula: see text] with J + J1 - 23J2 ≠ 0 has only one additional BS. (ii) The energy z of this additional BS is degenerate ν - 1 times. (iii) If Λ ≠ (Λ0;Λ0;…;Λ0), we show the existence no more 2ν + 1 bound states in the system in ν-dimensional lattice.

DEPHASING IN PRESENCE OF A MAGNETIC FIELD

Effect of the magnetic field on the rate of phase breaking is studied. It is shown that the magnetic field resulting in the decrease of phase relaxation rate [Formula: see text] makes the negative magnetoresistance due to suppression of the electron interference to be smoother in shape and lower in magnitude than that found with constant [Formula: see text]-value. Nevertheless our analysis shows that experimental magnetoconductance curves can be well fitted by the Hikami–Larkin–Nagaoka expression.1 The fitting procedure gives the value of τ/τϕ, where τ is the quasi-momentum relaxation time, which is close to the value of τ/τϕ(B = 0) with an accuracy of 25% or better when the temperature varies within the range from 0.4 to 10 K. The value of the prefactor α found from this procedure lies within the interval 0.9–1.2.