### The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles

Sergio Albeverio, Saidakhmat N. Lakaev, Tulkin H. Rasulov

Methods Funct. Anal. Topology **13** (2007), no. 1, 1-16

A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.

### On $*$-representations of algebras given by graphs

Methods Funct. Anal. Topology **13** (2007), no. 1, 17-27

*-algebras given by trees generated by projections with Temperley-Lieb type relations are considered in this work. Formulas of representations are built by algorithms for *-algebras, associated with the Dynkin diagrams, and we get estimates for the parameters at which non-trivial *-representations of the *-algebras exist.

### Spectral measure of commutative Jacobi field equipped with multiplication structure

Methods Funct. Anal. Topology **13** (2007), no. 1, 28-42

The article investigates properties of the spectral measure of the Jacobi field constructed over an abstract Hilbert rigging $H_-\supset H\supset L\supset H_+.$ Here $L$ is a real commutative Banach algebra that is dense in $H.$ It is shown that with certain restrictions, the Fourier transform of the spectral measure can be found in a similar way as it was done for the case of the Poisson field with the zero Hilbert space $L^2(\Delta,d u).$ Here $\Delta$ is a Hausdorff compact space and $ u$ is a probability measure defined on the Borel $\sigma$-algebra of subsets of $\Delta.$ The article contains a formula for the Fourier transform of a spectral measure of the Jacobi field that is constructed over the above-mentioned abstract rigging.

### On quadruples of linearly connected projections and transitive systems of subspaces

Yulia Moskaleva, Vasyl Ostrovskyi, Kostyantyn Yusenko

Methods Funct. Anal. Topology **13** (2007), no. 1, 43-49

We study conditions under which the images of irreducible quadruples of linearly connected projections give rise to all transitive systems of subspaces in a finite dimensional Hilbert space.

### Superstable criterion and superstable bounds for infinite range interaction I: two-body potentials

Methods Funct. Anal. Topology **13** (2007), no. 1, 50-61

A continuous infinite system of point particles interacting via two-body infinite-range potential is considered in the framework of classical statistical mecha ics. We propose some new criterion for interaction potentials to be superstable and give a very transparent proof of the Ruelle's uniform bounds for a family of finite volume correlation functions. It gives a possibility to prove that for any temperature and chemical activity there exists at least one Gibbs state. This article is a generalization of the work \cite{Re98} for the case of infinite range interaction potential.

### Classification of noncompact surfaces with boundary

A. O. Prishlyak, K. I. Mischenko

Methods Funct. Anal. Topology **13** (2007), no. 1, 62–66

We give a topological classification of noncompact surfaces with any number of boundary components.

### On models of function type for a special class of normal operators in Krein spaces and their polar representation

Methods Funct. Anal. Topology **13** (2007), no. 1, 67-82

The paper is devoted to a function model representation of a normal operator $N$ acting in a Krein space. We assume that $N$ and its adjoint operator $N^{\#}$ have a common invariant subspace $L_{+}$ which is a maximal nonnegative subspace and has a representation as a sum of a finite-dimensional neutral subspace and a uniformly positive subspace. For $N$ we construct a model representation as the multiplication operator by a scalar function acting in a suitable function space. This representation is applied to the problem of existence of a polar representation for normal operators of $D_{\kappa}^+$-class.

### On the Gauss-Manin connection in cyclic homology

Methods Funct. Anal. Topology **13** (2007), no. 1, 83-94

Getzler constructed a flat connection in the periodic cyclic homology, called the Gauss-Manin connection. In this paper we define this connection, and its monodromy, at the level of the periodic cyclic complex. The construction does not depend on an associator, and provides an explicit structure of a DG module over an auxiliary DG algebra. This paper is, to a large extent, an effort to clarify and streamline our work [4] with Yu.L. Daletsky.

### On Whitney constants for differentiable functions

Methods Funct. Anal. Topology **13** (2007), no. 1, 95-100

Some estimates of the constants in Whitney inequality for the differentiable functions are obtained.