Valentin A. Zagrebnov

On the Purity of the Limiting Gibbs State for the Ising Model on the Bethe Lattice

We give a proof that for the Ising model on the Bethe lattice, the limiting Gibbs state with zero effective field (disordered state) persists to be pure for temperature below the ferromagnetic critical temperatureTcF until the critical temperatureTcSG of the corresponding spin-glass model. This new proof revises the one proposed earlier.

The Bogoliubov model of weakly imperfect Bose gas

Abstract We present a systematic account of known rigorous results about the Bogoliubov model of weakly imperfect Bose gas (WIBG). This model is a basis of the celebrated Bogoliubov theory of superfluidity, although the physical phenomenon is, of course, more complicated than the model. The theory is based on two Bogoliubovs ansatze: the first truncates the full Hamiltonian of the interacting bosons to produce the WIBG, whereas the second substitutes some operators by c-numbers (the Bogoliubov approximation). After some historical remarks, and physical and mathematical motivations of this Bogoliubov treatment of the WIBG, we turn to revision of the Bogoliubovs ansatze from the point of view of rigorous quantum statistical mechanics. Since the exact calculation of the pressure and the behaviour of the Bose condensate in the WIBG are available, we review these results stressing the difference between them and the Bogliubov theory. One of the main features of the mathematical analysis of the WIBG is that it takes into account quantum fluctuations ignored by the second Bogoliubov ansatz. It is these fluctuations which are responsible for indirect attraction between bosons in the fundamental mode. The latter is the origin of a nonconventional Bose condensation in this mode, which has a dynamical nature. A (generalized) conventional Bose–Einstein condensation appears in the WIBG only in the second stage as a result of the standard mechanism of the total particle density saturation. It coexists with the nonconventional condensation. We give also a review of some models related to the WIBG and to the Bogoliubov theory, where a similar two-stage Bose condensation may take place. They indicate possibilities to go beyond the Bogoliubov theory and the Hamiltonian for the WIBG.

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization à la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.

The one-particle energy spectrum of weakly coupled quantum rotators

The ground state of a lattice model of weakly interacting quantum rigid rotators is analyzed by the cluster expansion method applied to its Feynman–Kac representation. The Hamiltonian of the infinite crystal in the ground state is shown to have a branch of absolutely continuous spectrum separated by gaps from the rest of the spectrum, describing the one-particle excitations.

DISORDERED BOSE EINSTEIN CONDENSATES WITH INTERACTION IN ONE DIMENSION

We study the effects of random scatterers on the ground state of the one-dimensional Lieb?Liniger model of interacting bosons on the unit interval in the Gross?Pitaevskii regime. We prove that Bose?Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wavefunction of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers and strong interactions the wavefunction extends over the whole interval. A high density of scatterers and weak interactions, on the other hand, lead to localization of the wavefunction in a fragmented subset of the interval.

Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas

We show that the pressure of the Bogoliubov weakly imperfect Bose gas (WIBG) can be calculated exactly in the thermodynamic limit. We point out the sufficient and necessary conditions for it not to equate with the pressure of the ideal Bose gas (IBG). We prove that they differ only in that part of the phase diagram where the WIBG has a Bose condensate. We show that in contast to the conventional Bose condensate (e.g. in the IBG) the condensate in the WIBG is due to an effective attraction between bosons in the zero-mode.

Long-range order in a lattice-gas model of nematic liquid crystals

A lattice model with full rotational symmetry is discussed for describing nematic liquid crystals. It is a lattice-gas approximation of continuum liquid with dispersion forces between long centrosymmetric molecules. Orientational long-range order at low temperatures and large chemical potentials is proven using the combination of an infrared bound and chessboard estimates.

A probabilistic approach to parallel dynamics for the Little-Hopfield model

Presents new results on a probabilistic approach to parallel dynamics of the Little-Hopfield model. The authors propose a truncated auxiliary dynamics method to control a feedback noise in this symmetrical neural network with full connection. It allows them to propose an ansatz for derivation of the explicit recurrence relations for the main and residual (noisy) overlap evolution for arbitrary discrete moment t.

A model with coexistence of two kinds of Bose condensation

We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.

ANOMALOUS ELECTRON TRAPPING BY LOCALIZED MAGNETIC FIELDS

We consider an electron with an anomalous magnetic moment g > 2 confined to a plane and interacting with a non-zero magnetic field B perpendicular to the plane. We show that if B has compact support and the magnetic flux in natural units is , the corresponding Pauli Hamiltonian has at least bound states, without making any assumptions about the field profile. Furthermore, in the zero-flux case there is a pair of bound states with opposite spin orientations. Using a Birman-Schwinger technique, we extend the last claim to a weak rotationally symmetric field with , thus correcting a recent result. Finally, we show that under mild regularity assumptions existence of the bound states can also be proved for non-symmetric fields with tails.