# Authors Index, Vol. 19, 2013

### Schrödinger operators with nonlocal potentials

Sergio Albeverio, Leonid Nizhnik

Methods Funct. Anal. Topology **19** (2013), no. 3, 199-210

We describe selfadjoint nonlocal boundary-value conditions for new exact solvable models of Schrödinger operators with nonlocal potentials. We also solve the direct and the inverse spectral problems on a bounded line segment, as well as the scattering problem on the whole axis for first order operators with a nonlocal potential.

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

Aleksandra Ananieva, Nataly Goloshchapova

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.

### Factorizations of nonnegative symmetric operators

Methods Funct. Anal. Topology **19** (2013), no. 3, 211-226

We prove that each closed denselydefined and nonnegative symmetric operator $\dot A$ having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form $\dot A=\mathcal L\mathcal L_0$, where $\mathcal L_0$ is aclosed nonnegative symmetric operator and $\mathcal L$ its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs $\mathcal L_0\subset\mathcal L$ ($\mathcal L_0$ isclosed and densely defined, $\mathcal L=\mathcal L^*\ge 0$) having the property${\rm dom\,}(\mathcal L\mathcal L_0)=\{0\}$ (and, in particular, ${\rm dom\,}(\mathcal L^2_0)=\{0\}$) is given.

### One-dimensional Schrödinger operators with general point interactions

Methods Funct. Anal. Topology **19** (2013), no. 1, 4-15

We consider various forms of boundary-value conditions for general one-dimensional Schrödinger operators with point interactions that include $\delta$-- and $\delta'$-- interactions, $\delta'$-- potential, and $\delta$-- magnetic potential. We give most simple spectral properties of such operators, and consider a possibility of finding their norm resolvent approximations.

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

Gerhard Bräunlich, Christian Hainzl, Robert Seiringer

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.

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

Alexei Daletskii, Ahsan Ul Haq

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 the range and kernel of Toeplitz and little Hankel operators

Namita Das, Pabitra Kumar Jena

Methods Funct. Anal. Topology **19** (2013), no. 1, 55-67

In this paper we study the interplay between the range and kernel of Toeplitz and little Hankel operators on the Bergman space. Let $T_\phi$ denote the Toeplitz operator on $L^2_a(\mathbb{D})$ with symbol $\phi \in L^\infty(\mathbb{D})$ and $S_\psi$ denote the little Hankel operator with symbol $\psi \in L^\infty(\mathbb{D}).$ We have shown that if ${\operatorname{Ran}} (T_\phi) \subseteq {\operatorname{Ran}} (S_\psi)$ then $\phi \equiv 0$ and find necessary and sufficient conditions for the existence of a positive bounded linear operator $X$ defined on the Bergman space such that $T_\phi X=S_\psi$ and ${\operatorname{Ran}} (S_\psi) \subseteq {\operatorname{{\operatorname{Ran}}}} (T_\phi).$ We also obtain necessary and sufficient conditions on $\psi \in L^\infty(\mathbb{D})$ such that ${\operatorname{Ran}} (T_\psi)$ is closed.

### The inner structure of the Jacobi-Laurent matrix related to the strong Hamburger moment problem

Methods Funct. Anal. Topology **19** (2013), no. 2, 97-107

The form of the Jacobi type matrix related to the strong Hamburger moment problem is known \cite{N5,BD}, i.e., there are known the zero elements of corresponding matrix. We describe the relations between of non-zero elements of such matrices, i.e., we describe ''the inner structure'' of the Jacobi-Laurent matrices related to the strong Hamburger moment problem.

### Myroslav Lvovych Gorbachuk (to his 75th birthday)

Methods Funct. Anal. Topology **19** (2013), no. 1, 1-3

### Fedor Semenovich Rofe-Beketov (to his 80th birthday)

Methods Funct. Anal. Topology **19** (2013), no. 3, 197-198

### An operator approach to Vlasov scaling for some models of spatial ecology

D. Finkelshtein, Yu. Kondratiev, O. Kutoviy

Methods Funct. Anal. Topology **19** (2013), no. 2, 108-126

We consider Vlasov-type scaling for Markov evolution of birth-and-death type in continuum, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. The existence of rescaled and limiting evolutions of correlation functions and convergence to the limiting evolution are shown. The obtained results enable us to derive a non-linear Vlasov-type equation for the density of the limiting system.

### On square root domains for non-self-adjoint Sturm-Liouville operators

Fritz Gesztesy, Steve Hofmann, Roger Nichols

Methods Funct. Anal. Topology **19** (2013), no. 3, 227-259

We determine square root domains for non-self-adjoint Sturm-Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb R$, the half-line $(a,\infty)$, $a \in \mathbb R$, or with the bounded interval $(a,b) \subset \mathbb R$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e. on $(a,b)$, we treat all separated boundary conditions at $a, b$.)

### Tachyon generalization for Lorentz transforms

Methods Funct. Anal. Topology **19** (2013), no. 2, 127-145

In the present paper we construct an expansion of the set of Lorentz transforms, which allows for the velocity of the reference frame to be greater than the speed of light. For maximum generality we investigate this tachyon expansion in the case of Minkowski space time over any real Hilbert space.

### Berezin number of operators and related questions

Mubariz T. Karaev, Nizameddin Sh. Iskenderov

Methods Funct. Anal. Topology **19** (2013), no. 1, 68-72

We prove some estimates for the Berezin number of operators on the reproducing kernel Hilbert spaces. We also give in terms of Berezin number necessary and sufficient conditions providing unitarity of invertible operator on the reproducing kernel Hilbert space. Moreover, we give a lower estimate for any operator on the Hardy space $H^{2}\left( \mathbb{D} \right) $ over the unit disc $\mathbb{D}.$

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

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.

### Spectral singularities of differential operator with triangular matrix coefficients

Methods Funct. Anal. Topology **19** (2013), no. 3, 260-267

For a non-selfadjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, we construct an example of such an operator having spectral singularities.

### Parabolic problems and interpolation with a function parameter

Valerii Los, Aleksandr A. Murach

Methods Funct. Anal. Topology **19** (2013), no. 2, 146-160

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce a refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem sets isomorphisms between appropriate spaces pertaining to this scale.

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

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 $A$.

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

Ie. Makedonskyi, A. Petravchuk

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.

### Schrödinger operators with complex singular potentials

Vladimir Mikhailets, Volodymyr Molyboga

Methods Funct. Anal. Topology **19** (2013), no. 1, 16-28

We study one-dimensional Schrödinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$. Particularly the class $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{{\operatorname{loc}}}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{{\operatorname{unif}}}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.

### Remarks on Schrödinger operators with singular matrix potentials

Vladimir Mikhailets, Volodymyr Molyboga

Methods Funct. Anal. Topology **19** (2013), no. 2, 161-167

In this paper, an asymmetric generalization of the Glazman-Povzner-Wienholtz theorem is proved for one-dimensional Schrödinger operators with strongly singular matrix potentials from the space $H_{loc}^{-1}(\mathbb{R}, \mathbb{C}^{m\times m})$. This result is new in the scalar case as well.

### On exit space extensions of symmetric operators with applications to first order symmetric systems

Methods Funct. Anal. Topology **19** (2013), no. 3, 268-292

Let $A$ be a symmetric linear relation with arbitrary deficiency indices. By using the conceptof the boundary triplet we describe exit space self-adjointextensions $\widetilde A^\tau$ of $A$ in terms of a boundary parameter $\tau$. We characterize certain geometrical properties of $\widetilde A^\tau$ and describe all $\widetilde A^\tau$ with ${\rm mul}\, \widetilde A^\tau=\{0\}$. Applying these results to general (possibly non-Hamiltonian) symmetric systems $Jy'- B(t)y=\Delta(t)y, \; t \in [a,b\rangle,$ we describe all matrix spectral functions of theminimally possible dimension such that the Parseval equality holdsfor any function $f\in L_\Delta^2([a,b \rangle)$.

### Parameter-elliptic operators on the extended Sobolev scale

Aleksandr A. Murach, Tetiana Zinchenko

Methods Funct. Anal. Topology **19** (2013), no. 1, 29-39

Parameter--elliptic pseudodifferential operators given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert--Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the absolute value of the parameter is large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.

### Indefinite moment problem as an abstract interpolation problem

Methods Funct. Anal. Topology **19** (2013), no. 2, 168-186

Indefinite moment problem was considered by M. G. Krein and H. Langer in 1979. In the present paper the general indefinite moment problem is associated with an abstract interpolation problem in generalized Nevanlinna classes. To prove the equivalence of these two problems we investigate the structure of de Branges space $H(m)$ associated with a generalized Nevanlinna function $m$. A general formula for description of the set of solutions of indefinite moment problem is found. It is shown that the Kein-Langer description can be derived from this formula by a special choice of biorthogonal system of polynomials.

### On stable $\mathcal{C}$-symmetries for a class of $\mathcal{PT}$-symmetric operators

Methods Funct. Anal. Topology **19** (2013), no. 1, 73-79

Recently, much attention is paid to the consideration of physical models described by $\mathcal{PT}$-symmetric Hamiltonians. In this paper, we establish a necessary and sufficient condition for existence of a stable $\mathcal{C}$-symmetry for a class of $\mathcal{PT}$-symmetric extensions of a symmetric operator $S$ with deficiency indices $(2,2)$.

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

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

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.

### Extended Weyl theorems and perturbations

Methods Funct. Anal. Topology **19** (2013), no. 1, 80-96

In this paper we study the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$, a variant of Weyl's type theorems introduced by Berkani. We established for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which the properties $(\rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ hold. Among other things, we study the stability of the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ for a polaroid operator $T$ acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with $T$.

### $\ell^{1}$-Munn ideal amenability of certain semigroup algebras

Methods Funct. Anal. Topology **19** (2013), no. 2, 187-190

In this paper we investigate ideal amenability of $\ell^{1}(G_{p})$, where $G_{p}$ is a maximal subgroup of inverse semigroup $S$ with uniformly locally finite idempotent. Also we find some conditions for ideal amenability of Rees matrix semigroup.

### Decomposition of a unitary scalar operator into a product of roots of the identity

Methods Funct. Anal. Topology **19** (2013), no. 2, 191-196

We prove that for all $m_1,m_2,m_3 \in \mathbb{N},~ \frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \leq 1$, every unitary scalar operator $\gamma I$ on a complex infinite-dimensional Hilbert space is a product $\gamma I = U_1 U_2 U_3$ where $U_i$ is a unitary operator such that $U_i^{m_i} = I$.

### The two-dimensional moment problem in a strip

Methods Funct. Anal. Topology **19** (2013), no. 1, 40-54

In this paper we study the two-dimensional moment problem in a strip $\Pi(R) = \{ (x_1,x_2)\in \mathbb{R}^2:\ |x_2| \leq R \}$, $R>0$. We obtained an analytic parametrization of all solutions of this moment problem. Usually the problem is reduced to an extension problem for a pair of commuting symmetric operators but we have no possibility to construct such extensions in larger spaces in a direct way. It turns out that we can find solutions without knowing the corresponding extensions in larger spaces. We used the fundamental results of Shtraus on generalized resolvents and some results from the measure theory.