This issue of Communications in Mathematical Analysis is devoted to various aspects of Analysis, Operator Theory and their applications in problems of Mathematical Physics. In particular, in this volume there are studied the compactness of commutators of convolution type operators with piecewise quasicontinuous data on weighted Lebesgue spaces with Muchkenhoupt weights, regularizers of Mellin pseudodifferential operators with slowly oscillating symbols of limited smoothness, the diffraction by a half-plane with different face impedances on an obstacle perpendicular to the boundary by using the theory of Wiener-Hopf and Hankel type operators on Bessel potential spaces. It is shown that the discrete series of irreducible unitary representation spaces of the non-compact group SO(2,1) can be naturally interpreted as discrete versions of the linear harmonic oscillator in standard non-relativistic quantum mechanics. Several papers in this volume are devoted to the structural analysis of Toeplitz operators with various specific classes of generating symbols and of the algebras generated by such Toeplitz operators. In particular, Toeplitz operators with quasi-radial symbols acting on the weighted Bergman space on the unit ball, Toeplitz operators with quasi-radial quasi-homogeneous symbols acting on the weighted pluriharmonic Bergman spaces on the unit ball, and Toeplitz operators with angular symbols acting on the Bergman spaces on the upper half-plane are studied. The Fredholm symbol algebras for the C-algebra generated by Toeplitz operators with piecewise quasicontinuous symbols and for the C-algebra generated by Toeplitz operators with piecewise continuous and slowly oscillating symbols, both acting on the Bergman space on the unit disk, are described.

Evolution of solitons is addressed in the framework of an extended nonlinear Schrödinger equation (NLSE), including a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term which is well known as an ingredient of the temporal-domain NLSE in optics. In the present context, it is induced by the underlying interaction of the high-frequency envelope wave with a damped lowfrequency wave mode. Also included are spatial inhomogeneity of both the second-order dispersion (SOD) and self-phase modulation (SPM). It is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, may be compensated by an upshift provided by the increasing SPM and SOD coefficients. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well.

We argue that the Meixner polynomials of a discrete variable are actually “encoded” within appropriate infinite-dimensional irreducible unitary representations of the three-dimensional Lorentz group SO(2,1). Hence discrete series of irreducible unitary representation spaces of the non compact group SO(2,1) can be naturally interpreted as discrete versions of the linear harmonic oscillator in standard non-relativistic quantum mechanics.

It is known that the Maxwell-Lorentz equations with Abraham’s rotating extended electron can be derived from the the Hamilton least action principle applying the variational Poincaré equations on the Lie group SO(3). We prove that, rewritten in the Euler angles, these equations imply the standard Euler-Lagrange and Hamiltonian equations.

In this paper a porous 3D system wherein the material porosity forms a percolating spanning cluster is studied. It is considered that the light nanoemitters are randomly incorporated into such a cluster and are pumped to the exited state by an external light beam. To find the optimal optical path between the radiated nanoemitters the Fermat principle was used and it is implemented by means of well-known the travelling salesman (TSP) approach generalized to 3D geometry. With the use of undirected graph mapping the TSP approach is applied for optimizing optical path (for principle of Fermat) to calculate the optimal light paths in such a percolation structure. The convergence of the algorithm for 3D incipient cluster is studied with details to exploit the space of feasible solutions when a collective light radiation of the internal nanoemitters becomes. Finally, the field correlations of the light emission that penetrates the entire 3D percolation cluster is discussed.

The paper is devoted to study classes of plane wave diffraction problems by a region which involves a crack with impedance boundary conditions. Conditions on the wave number and impedance parameters are found to ensure the well-posedness of the problems in a scale of Bessel potential spaces. Under such conditions, representations of the solutions are also obtained upon the consideration of some associated operators which, in a sense, combine operators of Wiener-Hopf and Hankel type.

In "Multiplicative Structure of the Resolvent Matrix for the Truncated Matricial Hausdorff Moment Problem", Operator Theory: Advances and Applications, (2012) by the author, a multiplicative decomposition of resolvent matrix $U^{(2n)}$ for the truncated Hausdorff matrix moment (THMM) problem via Blaschke–Potapov factors $b^{(2 j)}$ was obtained. In this work we show that every such Blaschke-Potapov factor can be represented as a product of block tridiagonal matrices containing Stieltjes matrix parameters (SMP) depending on a or b. This SMP are in turn a generalization of the Yu. Dyukarev’s Stieltjes parameters introduced in “Indeterminacy criteria for the Stieltjes matrix moment problem”, Mathematical Notes (2004).

We obtain multiplicative decompositions of the Blaschke–Potapov factors of the truncated Hausdorff matrix moment (THMM) problem in the case of an even number of moments. Our result develops the multiplicative representation of the resolvent matrix of the THMM problem in the case of even number of moments by I. Serikova in "The multiplicative structure of resolvent matrix of the moment problem on the kompact interval (case of even numbers of moments)", Vestnik Kharkov Univ. Ser. Mat. Prikl. Mat. i Mekh. no. 790 (2007). We show that every such Blaschke-Potapov factor can be represented as a product of tridiagonal block matrices containing Stieltjes matrix parameters (SMP) depending on a and b. These SMP in turn area generalization of the Dyukarev’s Stieltjes parameters introduced in ”Indeterminacy criteria for the Stieltjes matrix moment problem,” Mathematical Notes, Vol. 75 (2004).

An inverse spectral problem is studied for Sturm-Liouville differential operators on hedgehog-type graphs with generalized matching conditions in the interior vertices and with Dirichlet boundary conditions in the boundary vertices. A uniqueness theorem of recovering potentials from given spectral characteristics is provided, and a constructive solution for the inverse problem is obtained.

In this paper we obtain an algorithm for the truncated matrix Hausdorff moment problem with an odd number of given moments. The coefficients of the corresponding linear fractional matrix transformation can be calculated using the prescribed moments. No conditions besides solvability are assumed for the moment problem. The question of the determinateness of the moment problem is answered by (a part of) the algorithm as well. Several examples are provided.

Applying the theory of Calderon-Zygmund operators, we study the compactness of the commutators $[aI,W^0(b)]$ of multiplication operators $aI$ and convolution operators $W^0(b)$ on weighted Lebesgue spaces $L^p({\mathbb R},w)$ with $p\in(1,\infty)$ and Muckenhoupt weights $w$ for some classes of piecewise quasicontinuous functions $a\in PQC$ and $b\in PQC_{p,w}$ on the real line ${\mathbb R}$. Then we study two $C^*$-algebras $Z_1$ and $Z_2$ generated by the operators $aW^0(b)$, where $a,b$ are piecewise quasicontinuous functions admitting slowly oscillating discontinuities at $\infty$ or, respectively, quasicontinuous functions on ${\mathbb R}$ admitting piecewise slowly oscillating discontinuities at $\infty$. We describe the maximal ideal spaces and the Gelfand transforms for the commutative quotient $C^*$-algebras $Z_i^\pi=Z_i/{\mathcal K}$ $(i=1,2)$ where ${\mathcal K}$ is the ideal of compact operators on the space $L^2({\mathbb R})$, and establish the Fredholm criteria for the operators $A\in Z_i$.

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend only on the argument of the variable. This algebra is known to be commutative, and it is isometrically isomorphic to a certain algebra of bounded complex valued functions on the real numbers. In the paper we prove that the latter algebra consists of all bounded functions f that are very slowly oscillating on the real line in the sense that the composition of f with sinh is uniformly continuous with respect to the usual metric.

By means of the variational approximation (VA) and systematic simulations, we study dynamics and stability boundaries for fundamental solitons in a two-dimensional (2D) self-attracting Bose-Einstein condensate (BEC), trapped in an quasiperiodic optical lattice (OL), with the amplitude subject to periodic time modulation (the modulation frequency, ω. Regions of stability of the solitons against the collapse and decay are identified in the space of the model’s parameters.

We study the so-called quasi-radial operators, i.e., the operators that are invariant under the subgroup of the unitary group ${\mathfrak U}(n)$ formed by the block-diagonal matrices with unitary blocks of fixed dimensions. The quasi-radial Toeplitz operators appear naturally and play a crucial role under the study of the commutative Banach (not $C^*$) algebras of Toeplitz operators [1, 8]. They form an intermediate class of operators between the Toeplitz operators with radial $a=a(r)$, $r=\sqrt{|z_1|^2 + \ldots + |z_n|^2}$, and separately-radial $a = a(|z_1|, \ldots, |z_n|)$ symbols.

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ of slowly oscillating functions of limited smoothness introduced in [12]. We show that if ${\mathcal a} \in\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ does not degenerate on the ``boundary" of ${\mathbb R}_+\times {\mathbb R}$ in a certain sense, then the Mellin PDO $Op({\mathcal a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form $Op({\mathcal b})+K$ where $K$ is a compact operator on $L^p$ and ${\mathcal b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ such that ${\mathcal b}=1/{\mathcal a}$ on the ``boundary" of ${\mathbb R}_+\times {\mathbb R}$. This result complements the known Fredholm criterion from [12] for Mellin PDO's with symbols in the closure of $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ and extends the corresponding result by V.S. Rabinovich (see [16]) on Mellin PDO's with slowly oscillating symbols in $C^\infty({\mathbb R}_+\times {\mathbb R})$.

We derive dispersion estimates for solutions of the one-dimensional discrete wave equations. In particular, we weaken the conditions on the potentials of previous works.

The paper deals with the global robust feedback syntheses of a bounded control for a disturbed canonical system with an unknown bounded perturbation. Our approach is based on the controllability function method created by V. I. Korobov in 1979. We find a segment where the perturbation can vary and give a positional control which is independent of the perturbation and steers any initial point to the origin for any admissible perturbation from this segment. An estimate for the time of motion is given.

We describe the symbol (Calkin) algebra for the algebra generated by Toeplitz operators with piecewise continuous and slowly oscillating radial symbols that act on the Bergman space on the unit disk.

We study Toeplitz operators acting on the weighted pluriharmonic Bergman space on the unit ball, denoted here by $b_{λ}^2({\mathbb B}^n)$. We prove that, besides the C*-algebra generated by Toeplitz operators with radial symbols, there are commutative Banach algebras generated by Toeplitz operators. These Banach algebras are generated by Toeplitz operators with k-quasi-radial and k-quasi-homogeneous symbols.

We report results of systematic analytic and numerical study of the electromagnetic field generated by a moving with the uniform velocity source through a dispersive lossy metamaterial. The Cherenkov radiation in a far zone is considered with the use of 2D stationary phase method. In our simulations the Drude model is implemented for a metamaterial. As a result of passage of modulated charged source in metamaterial a double branching of the Cherenkov cone with negative refractive index is registered.

In the paper, $QC$ stands for the $C^*$-algebra of quasicontinuous functions on $\partial {\mathbb D}$ defined by D. Sarason in [10]. For a fixed subset $\Lambda:=\{ \lambda_1, \lambda_2, \dots, \lambda_n\}$ of the unit circle $\partial {\mathbb D}$, we define the algebra $PC$ of piecewise continuous functions in $\partial {\mathbb D} \setminus \Lambda$ with one-sided limits at each point $\lambda_k \in \Lambda$. We define $PQC$ as the $C^*$-algebra generated by both $PC$ and $QC$.

$\mathcal{A}^2({\mathbb D})$ stands for the Bergman space of the unit disk ${\mathbb D}$, that is, the space of square integrable and analytic functions defined on ${\mathbb D}$. Let ${\mathcal K} $ denote the ideal of compact operators acting on $\mathcal{A}^2({\mathbb D})$. Our goal is to describe the Calkin algebra ${\mathcal T}_{PQC}/ {\mathcal K}$, where ${\mathcal T}_{PQC}$ is the $C^*$-algebra generated by Toeplitz operators acting on $\mathcal{A}^2({\mathbb D})$ whose symbols are certain extensions of functions in $PQC$. A function defined on $\partial {\mathbb D}$ can be extended to the disk in many ways, the more natural extensions are the harmonic and the radial ones. In the final part of this paper we prove that the description of ${\mathcal T}_{PQC}$ does not depend on the extension chosen.

Neutral type systems considered in infinite-dimensional Hilbert space are analyzed for exact controllability characterization. The approach is based on the problem of moments using a Riesz basis of eigenvectors. The duality with observability is investigated. A criterion of exact observability is deduced.

The transmission and reflection coefficients for the scattering of a particle on one-dimensional potential are calculated by means of Spectral Parameter Power Series (SPPS). The results were compared with known results.

We present a survey of recent developments on a parabolic version of uniform rectifiability and parabolic singular integrals. In particular we describe some ideas to prove the equivalence between the parabolic uniform rectifiability of a set $E$ and the $L^2(E)$ boundedness of a class of Calder´on Zygmund integrals of parabolic type. We also describe a result on compactness of certain parabolic singular integrals, as well as some related open problems and conjectures.

We prove that the system of equations describing waves in a two-layer fluid over a rectangular barrier of small height possesses embedded trapped modes (eigenvalues submerged in the continuous spectrum) for certain values of the width of the barrier and that these eigenvalues are analytic in the small parameter characterizing the height of the barrier. We do this by means of purely elementary considerations constructing explicit solutions and thus confirm the results of [7] obtained for general perturbations of the depth of the fluid in the particular case of a rectangular barrier.

The barotropic vorticity equation describing the vortex dynamics of viscous and forced incompressible fluid on a rotating sphere is considered. This equation is also used for studying the large-scale dynamics of barotropic atmosphere. Operators of orthogonal projection on the subspaces of homogeneous spherical polynomials and derivatives of real order for functions are introduced. A family of Hilbert spaces of generalized functions having fractional derivatives of real order s is introduced, and a few embedding theorems are given. An equation for the evolution of kinetic energy of perturbations to a basic flow is analyzed. A relationship between the rate of generation of kinetic energy perturbations and the eigenfunctions of the symmetric part of the operator linearized about the basic flow is shown. As an illustrative example, the numerical solution of the spectral problem for such operator is discussed in the case when the basic flow is the climatic January circulation.

We give the complete description of nonlinear control systems of the class $C^1$ with multi-dimensional control that are linearizable by means of changes of variables of the class $C^2$.