Leading Branches of the Transfer-Matrix Spectrum for Lattice Spin Systems (Quasi-Particles of Different Species)

For lattice systems under high temperatures T with compact or finite spin we construct three invariant subspaces of the transfer-matrix, which can be interpreted as the spaces of states for quasi-particles of two different species and the space of states for two particles of the first species. We formulate a condition on a priori distribution guaranteeing that the spectrum of the transfer-matrix on these subspaces are not overlapping.     

Invariant subspaces of clustering operators. II

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

The limiting theorems for a random walk of two particles on the lattice ℤν

It is shown that the difference between the probability distributions of the particles positions at time t as t for homogeneous and inhomogeneous random walk of two particles on the lattice Z ³ has an order (>0 is a constant), if the distance |z| between the particles is large enough. As a consequence the integral limit theorem was proved in this case.partially supported by Russian Fund of Fundamental Research 93-011-1470.     

Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model

We find the asymptotic decrease of correlations < _{A +y} , _B >,yZ ^v +1, |y|, in the Ising model at high temperatures. For the case when monomials _A and _B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension . For even monomials _A , _B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) =1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary . Here we use besides the saddle-point method, some arguments from scattering theory.     

A new method in the scattering theory

A new method allowing one to establish the existence of wave operators is demonstrated on the example of the well-known Friedrichs’ model.     

Estimates for the interaction potential between two atoms

We consider a quantum system consisting of two electrons in the Coulomb field of two immovable nuclei spaced at a fixed distance, which is considered the problem parameter. We use spectral theory methods to obtain estimates of the system ground-state energy as a function of the internuclear distance. We obtain part of the results under the assumption that the Coulomb interaction constant between different atoms is small. We discuss the importance of the presented investigation for quantum chemistry.