R. A. Minlos

A QUANTUM CRYSTAL MODEL IN THE LIGHT-MASS LIMIT: GIBBS STATES

Ground and temperature quantum Gibbs states are constructed for a ferroelectric anharmonic quantum oscillator model with small masses. It is shown that they possess mixing properties. The construction relies on the Feynman–Kac–Nelson representation of the conditional reduced density matrices and on the cluster expansions for the corresponding Gibbs field of trajectories.

GROUND STATE PROPERTIES OF THE NELSON HAMILTONIAN: A GIBBS MEASURE-BASED APPROACH

The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.

Invariant subspaces of clustering operators

A class of clustering operators is defined which is a generalization of a transfer matrix of a Gibbs lattice field with an exponential decay of correlations. It is proved that for small values ofβ the clustering operator has invariant subspaces which are similar tok-particle subspaces of the Fock space. The restriction of the clustering operator onto these subspaces resembles the operator exp(-Hk, whereHk is thek- particle Schrödinger Hamiltonian in nonrelativistic quantum mechanics. The spectrum of eachHk,k⩾1, is contained in the interval (C1βk,C2βk). These intervals do not intersect with each other.

Gibbs Measures for Brownian Paths Under the Effect of an External and a Small Pair Potential

We consider Brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a Gibbs measure. Uniqueness of the Gibbs measure for two cases is shown. Also the typical path behaviour, the degree of mixing and some further properties are derived. We use cluster expansion in the small coupling parameter.

ONE-PARTICLE SUBSPACE OF THE GLAUBER DYNAMICS GENERATOR FOR CONTINUOUS PARTICLE SYSTEMS

We consider a Glauber-type stochastic dynamics of continuous particle systems in ℝd. We construct a one-particle invariant subspace of the generator of this dynamics in the high temperature and low density regime. We prove that under some additional assumptions on the decay of the potential the restriction of the generator on the one-particle subspace is unitary equivalent to the operator of the multiplication by a bounded smooth real-valued function. As a consequence we estimate the spectral gap of the generator and find the second gap between the one-particle branch and the rest of the spectrum.

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.

Spectrum and scattering in a “spin-boson” model with not more than three photons

A model of radiative decay with a fixed atom and not more than three photons is studied. A spectral analysis of the Hamiltonian is made. This is done by means of scattering theory in a pair of spaces with a specially chosen embedding. The existence of wave operators and their asymptotic completeness are proved. The constructions are based on a detailed analysis of the resolvent of the Hamiltonian.

Invariant subspaces of clustering operators. II

Clustering operators, when restricted tok-particle invariant subspaces, are shown still to cluster.

LOWER SPECTRAL BRANCHES OF A PARTICLE COUPLED TO A BOSE FIELD

The structure of the lower part (i.e. e-away below the two-boson threshold) spectrum of Frohlichs polaron Hamiltonian in the weak coupling regime is obtained in spatial dimension d ≥ 3. It contains a single polaron branch defined for total momentum p ∈ G(0), where G(0) ⊂ ℝd is a bounded domain, and, for any p ∈ ℝd, a manifold of polaron + one-boson states with boson momentum q in a bounded domain depending on p. The polaron becomes unstable and dissolves into the one-boson manifold at the boundary of G(0). The dispersion laws and generalized eigenfunctions are calculated.

On Pointlike Interaction between Three Particles: Two Fermions and Another Particle

The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with mass 𝑚>0) was studied in many works. In most of these works, a family of one-parametric symmetrical operators {𝐻𝜀,𝜀∈ℝ1} is considered as such Hamiltonians. In addition, the question about the self-adjointness of 𝐻𝜀 is equivalent to the one concerning the self-adjointness of some auxiliary operators {𝒯𝑙,𝑙=0,1,…} acting in the space 𝐿2(ℝ1