# Vol. 19 (2013), no. 4

### Logarithmic Sobolev inequality for a class of measures on configuration spaces

Methods Funct. Anal. Topology 19 (2013), no. 4, 293-300

We study a class of measures on the space $\Gamma _{X}$ of locally finiteconfi\-gurations in $X=\mathbb{R}^{d}$, obtained as images of ''lattice'' Gibbs measures on $X^{\mathbb{Z}^{d}}$ with respect to an embedding $\mathbb{Z}^{d}\subset \mathbb{R}^{d}$. For these measures, we prove the integration by parts formula andlog-Sobolev inequality.

### On Kondratiev spaces of test functions in the non-Gaussian infinite-dimensional analysis

N. A. Kachanovsky

Methods Funct. Anal. Topology 19 (2013), no. 4, 301-309

A blanket version of the non-Gaussian analysis under the so-called bior hogo al approach uses the Kondratiev spaces of test functions with orthogonal bases given by a generating function $Q\times H \ni (x,\lambda)\mapsto h(x;\lambda)\in\mathbb C$, where $Q$ is a metric space, $H$ is some complex Hilbert space, $h$ satisfies certain assumptions (in particular, $h(\cdot;\lambda)$ is a continuous function, $h(x;\cdot)$ is a holomorphic at zero function). In this paper we consider the construction of the Kondratiev spaces of test functions with orthogonal bases given by a generating function $\gamma(\lambda)h(x;\alpha(\lambda))$, where $\gamma :H\to\mathbb C$ and $\alpha :H\to H$ are holomorphic at zero functions, and study some properties of these spaces. The results of the paper give a possibility to extend an area of possible applications of the above mentioned theory.

### On the extremal extensions of a non-negative Jacobi operator

Methods Funct. Anal. Topology 19 (2013), no. 4, 310-318

We consider the minimal non-negative Jacobi operator with $p\times p-$matrix entries. Using the technique of boundary triplets and the corresponding Weyl functions, we describe the Friedrichs and Krein extensions of the minimal Jacobi operator. Moreover, we parametrize the set of all non-negative extensions in terms of boundary conditions.

### Spectral functions of the simplest even order ordinary differential operator

Anton Lunyov

Methods Funct. Anal. Topology 19 (2013), no. 4, 319-326

We consider the minimal differential operator $A$ generated in $L^2(0,\infty)$ by the differential expression $l(y) = (-1)^n y^{(2n)}$. Using the technique of boundary triplets and the corresponding Weyl functions, we find explicit form of the characteristic matrix and the corresponding spectral function for the Friedrichs and Krein extensions of the operator $A$.

### Spectral properties of Sturm-Liouville equations with singular energy-dependent potentials

Nataliya Pronska

Methods Funct. Anal. Topology 19 (2013), no. 4, 327-345

We study spectral properties of energy-dependent Sturm--Liouville equations, introduce the notion of norming constants and establish their interrelation with the spectra. One of the main tools is the linearization of the problem in a suitable Pontryagin space.

### On inverse spectral problems for self-adjoint Dirac operators with general boundary conditions

D. V. Puyda

Methods Funct. Anal. Topology 19 (2013), no. 4, 346-363

We consider the self-adjoint Dirac operators on a finite interval with summable matrix-valued potentials and general boundary conditions. For such operators, we study the inverse problem of reconstructing the potential and the boundary conditions of the operator from its eigenvalues and suitably defined norming matrices.

### On contact interactions as limits of short-range potentials

Methods Funct. Anal. Topology 19 (2013), no. 4, 364-375

We reconsider the norm resolvent limit of $-\Delta + V_\ell$ with $V_\ell$ tending to a point interaction in three dimensions. We are mainly interested in potentials $V_\ell$ modelling short range interactions of cold atomic gases. In order to ensure stability the interaction $V_\ell$ is required to have a strong repulsive core, such that $\lim_{\ell \to 0} \int V_\ell >0$. This situation is not covered in the previous literature.

### On finite dimensional Lie algebras of planar vector fields with rational coefficients

Methods Funct. Anal. Topology 19 (2013), no. 4, 376-388

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as subalgebras of this algebra.