# Vladimir RabinovichInstituto Politécnico Nacional | IPN · Escuela Superior de Ingeniería Mecánica y Eléctrica

Vladimir Rabinovich

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232

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Introduction

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## Publications

Publications (232)

We consider the Dirac operators with singular potentials $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$
(1)
where $$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m}= \sum \limits _{j=1}^{n} \alpha _{j}\left( -i\partial _{x_{j}}+A_{j}\right) +\alpha _{n+...

In this work, we consider the one‐dimensional Dirac operator DQ+Qsu(x)=1iσ2ddx+Q(x)+Qs(x)u(x),$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s}u(x)=\left(\frac{1}{\mathrm{i}}{\sigma}_2\frac{d}{dx}+\mathrm{Q}(x)+{\mathrm{Q}}_s(x)\right)u(x), $$ where σ2$$ {\sigma}_2 $$ is Pauli's matrix, Q$$ \mathrm{Q} $$ is a 2×2$$ 2\times 2 $$‐matrix representing a re...

We consider the Dirac operators with singular potentials $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A} ,\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$
(1)
where $$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m}= {\displaystyle \sum \limits _{j=1}^{n}} \alpha _{j}\left( -i\partial _{x_{j}}+A_{j}\ri...

The work is devoted to the asymptotic and numerical analysis of the wave function propagating in two-dimensional quantum waveguides with confining potentials supported on slowly varying tubes. The leading term of the asymptotics of the wave function is determined by an adiabatic approach and the WKB approximation. Unlike other similar studies, in t...

We consider the magnetic anisotropic Schrödinger operator on ℝ n
H ϱ , a , W u ( x ) = ( D + a ( x ) ) · ϱ ( x ) ( D + a ( x ) ) + W ( x ) u ( x ) , x ∈ ℝ n , (1)
where ϱ ( x ) = ( ϱ i j ( x ) ) i , j = 1 n , D = − i ∇ , a ( x ) = ( a 1 ( x ) , … , a n ( x ) ) is the vector potential of the magnetic field and W(x) is the scalar potential of the ele...

The decision-making process of positioning a new well must include a comprehensive analysis of the target rock properties to be drilled. This analysis is based on the knowledge acquired from vintage wells and any previous studies done for earlier operators in the area or nearby areas. Some concepts like brittleness, porosity and organic matter may...

We consider the 3-D Dirac operator with variable regular magnetic and electrostatic potentials, and singular potentials (1) D A , Φ , Q sin u ( x ) = D A , Φ + Q sin u ( x ) , x ∈ R 3 (1) where (2) D A , Φ = ∑ j = 1 3 α j i ∂ x j + A j ( x ) + α 0 m + Φ ( x ) I 4 , (2) Q sin = Γ ( s ) δ Σ is the singular potential with Γ ( s ) = Γ i j ( s ) i , j =...

We consider the $3-D$ Dirac operator $\mathfrak{D}_{\boldsymbol{A},\Phi ,Q_{\sin }}$ with variable regular magnetic and electrostatic potentials $ \boldsymbol{A}$,$\Phi $ and with singular potentials $Q_{\sin }$ with support on a smooth unbounded surface $\Sigma \subset \mathbb{R}^{3}$ which divides $\mathbb{R}^{3}$ on two open domains $\Omega_{\pm...

We consider operators of boundary value problems for 3D- Dirac operators in unbounded domains with the uniformly regular boundary. We give effective conditions of self-adjointness of operators under consideration and a description of their essential spectra. We also give applications to operators of the MIT bag problems for unbounded domains

We consider the Dirac operator on ℝ of the form 𝔇u(x)=Jddx+Q+Qsu(x),x∈ℝ where J=0−110,Q(x)=p(x)+r(x)q(x)q(x)−p(x)+r(x),p,q,r∈L∞(ℝ) is the regular potential, and 1Qs(x)=∑y∈𝕐Γyδ(x−y) is the singular potential, δ is the Dirac delta-function, Γy=(γij(y))i,j=1,2 is a 2 × 2-matrix with elements γij(y)∈l∞(𝕐),i,j=1,2,𝕐⊂ℝ is an infinite or finite discrete s...

We consider the Dirac operator on ℝ ³ , D A , Φ , Q s = ∑ j = 1 3 α j ( i ∂ x j + A j ( x ) ) + α 0 m − Φ ( x ) I 4 + Q s , with magnetic potential A ( x ) = ( A 1 ( x ), A 2 ( x ), A 3 ( x ))) and electrostatic potential Φ( x ), where α j ,j = 0, 1, 2, 3 are the Dirac matrices, Q s = Γ δ S is a singular potential where δ S is the Dirac δ —function...

In this paper we consider a quantum waveguide that consists of three strata ∏ 0 = {(x, x 3 ) ∈ ℝ ³ : x 3 < 0}, ∏ 0, h = {(x, x 3 ) ∈ ℝ ³ : 0 < x 3 < h }, ∏ h = {(x, x 3 ) ∈ ℝ ³ : x 3 > h }, where x = ( x 1 , x 2 ) ∈ ℝ ² . A potential of the form q = q r + q s is established in this structure, where q r is a regular bounded potential depending on on...

In this work the Zakharov-Shabat system is addressed to obtain a pair of supersymmetric Schrödinger equations. The scattering and resonance states of these equations are investigated. Explicit solutions for the equations are obtained in the form of power series of the spectral parameter. In the case of the scattering states, we obtain expressions f...

We describe the location of the essential spectrum of one-dimensional Dirac operators with singular potentials supported on discrete sets in ℝ in terms of limit operators.

В заметке дается описание существенного спектра одномерных операторов Дирака с сингулярными потенциалами, имеющими носители на дискретных множествах в $\mathbb{R}$, в терминах предельных операторов.

We consider Schrödinger operators H = −Δ + W + Ws on with regular potentials and singular potentials with supports on unbounded enough smooth hypersurfaces Γ. In particular, we consider singular potentials that are linear combinations of δ−functions on Γ and its normal derivatives. We consider extensions of H as symmetric operators in with domain t...

In this paper, we consider one‐dimensional Schrödinger operators Sq on with a bounded potential q supported on the segment and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in defined by the Schrödinger operator and matrix point conditions at the ends h0, h1. By using the spectral parameter power ser...

The paper is devoted to parameter dependent differential operators on graphs with general vertex conditions. We prove that if the differential operators are elliptic with parameter and an analogue of the well-known Lopatinsky condition at the vertices is satisfied, then the parameter dependent problem on the graph is invertible for large values of...

The paper is devoted to the spectral properties of one-dimensional Schrödinger operators
(1)Squ(x)=(-d2dx2+q(x))u(x), x∈ℝ,
with potentials q = q0+qs, where q0∈L∞(ℝ) is a regular potential, and qs∈D′(ℝ) is a singular potential with support on a discrete infinite set Y⊂ℝ. We consider the extension H of formal operator (1) to an unbounded operator in...

Let Γ be an unbounded metric-oriented graph embedded in Rn, E, V be countable sets of edges e∈E, and vertices v∈V of Γ. The graph Γ is equipped with a differential equation (1) Au(x)=∑j=0raj(x)u(j)(x)=f(x),x∈Γ∖V(1) with piece-wise smooth coefficients aj, j=,0,1,…,r, and general conditions at the vertices v∈V (2) Bku(v)=∑j=0mkbj,k(v)uEv(j)(v)=ϕk(v)∈...

The main aim of the paper is the study of the Fredholm property and essential spectra of electromagnetic Schrödinger operators \(\mathcal{H}\) on graphs periodic with respect to a group \(\mathbb{G}\) isomorphic to ℤk. We consider the Schrödinger operators with nonperiodic electric and magnetic potentials and with general nonperiodic conditions on...

We consider pseudodifferential operators of variable order acting on Besov spaces of variable smoothness. We prove the boundedness and compactness of such operators and study the Fredholm property of pseudodifferential operators of variable order with symbols slowly oscillating at infinity on weighted Besov spaces with variable smoothness.

We give a description of the essential spectra of unbounded operators ℋq on L²(Γ) determined by the Schrödinger operators −d²/dx² + q(x) on the edges of Γ and general vertex conditions. We introduce a set of limit operators of ℋq such that the essential spectrum of ℋq is the union of the spectra of limit operators. We apply this result to describe...

We consider pseudodifferential operators of variable orders acting
in Besov spaces of variable smoothness. We prove the boundedness
and compactness of operators under consideration and study the
Fredholm property of pseudodifferential operators of variable
orders with slowly oscillating at infinity symbols in weighted
Besov spaces with variable smo...

The paper is devoted to the Fredholm theory of differential operators on ifinite metric graphs imbedded in ℝⁿ and periodic with respect to a group G isomorphic to Zr, 1 ≤ r ≤ n. It is assumed that the coefficients of differential operators and vertex conditions are not periodic. Necessary and sufficient conditions for operators under consideration...

In this paper, we consider periodic metric graphs embedded in , equipped by Schrödinger operators with bounded potentials q, and -type vertex conditions. Graphs are periodic with respect to a group isomorphic to . Applying the limit operators method, we give a formula for the essential spectra of associated unbounded operators consisting of a union...

Let Γ be a simply connected unbounded C2-hypersurface in ℝn such that Γ divides ℝn into two unbounded domains D±. We consider the essential spectrum of Schrödinger operators on ℝn with surface δΓ-interactions which can be written formally as $${H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}$$, where −Δ is the nonnegative Laplacia...

Let \(\Gamma \subset \mathbb {R}^{n}\) be a graph periodic with respect to the action of a group \(\mathbb {G}\) isomorphic to \(\mathbb {Z}^{m},1\le m\le n. \) We consider a one-dimensional Schrödinger operator $$\begin{aligned} S_{q}u(x)=\left( -\frac{d^{2}}{dx^{2}}+q(x)\right) u(x),u\in C_{0}^{\infty }(\Gamma \backslash \mathcal {V)},q\in L^{\in...

We consider the electron propagation in a cylindrical quantum waveguide where D is a bounded domain in described by the Dirichlet problem for the Schrödinger operator
where x=(x1, x2), , is the transversal confinement potential, and is the impurity potential. We construct the left and right transition matrices and give an numerical algorithm for t...

We present in the paper an effective numerical method for the determination of the spectra of periodic metric graphs equipped by Schrödinger operators with real-valued periodic electric potentials as Hamiltonians and with Kirchhoff and Neumann conditions at the vertices. Our method is based on the spectral parameter power series method, which leads...

We consider pseudodifferential operators of variable orders acting in Hölder–Zygmund spaces of variable smoothness. We prove the boundedness and compactness of the operators under consideration and study the Fredholm property of pseudodifferential operators with slowly oscillating at infinity symbols in the weighted Hölder–Zygmund spaces of variabl...

The paper is devoted to the method of potential operators for boundary and transmission problems in domains in \(\mathbb{R}^n\) with smooth unbounded boundaries for the anisotropic Helmholtz operators $$\mathcal{H}u(x)=\bigtriangledown \cdot a(x)\bigtriangledown u(x)+b(x)u(x),\quad x\in \mathbb{R}^N$$ with variable coefficients, where \(a\;=\;(a^{k...

The paper is devoted to the Lp-theory of boundary integral operators for boundary value problems described by anisotropic Helmholtz operators with variable coefficients in unbounded domains with unbounded smooth boundary. We prove the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Bessel-potential spaces Hs,p...

We investigate the underwater acoustic field in the stratified ocean generated by moving in the air sources. We obtain asymptotic formulas expressed in terms of the retarded time and the Doppler-shifted frequency. The spectral parameter power series method is implemented to find the wave numbers of the propagating modes, their group velocity and an...

The underwater acoustic wave propagation in the stratified ocean produced by moving airborne sources is investigated. We construct the asymptotics of the acoustic pressure in the ocean on large horizontal distances between a moving in the air source and a receiver located in the ocean. Our study is based on the representation of the acoustic field...

In the present work, we consider a modulated point source in an arbitrary motion in an isotropic planarly layered waveguide. The radiation field generated by this source is represented in the form of double oscillatory integrals in terms of the time and the frequency, depending on the large parameter λ. By means of the stationary phase method, we a...

In the present work, we consider an isotropic planarly layered waveguide in the case of polarized waves, and calculate the Green’s function associated with a motionless unit point source in the stationary state. On the basis of the Green’s function, we approach the dynamical problem of a point source that is in uniform motion in the waveguide. The...

We consider the fields of the acoustic pressure in the stratified ocean produced by the non-uniformly moving sources. We obtain the asymptotic formulas for the far field taking into account the mode and lateral wave contributions. We propose the spectral parameter power series method (SPPS method) of the dispersion equation applicable for the calcu...

The paper is devoted to the boundary integral equations method for the diffraction problems on obstacles D in \({\mathbb{R}^{n}}\) with smooth unbounded boundaries for Helmholtz operators with variable coefficients. The diffraction problems are described by the Helmholtz operators
$$\mathcal{H}u(x)=\left( \rho (x)\nabla \cdot \rho ^{-1}(x)\nabla+a(...

We consider a general boundary value problem1 in a smooth unbounded domain with conical exits at infinity, where the coefficients belong to the space of infinitely differentiable functions on bounded together with all derivatives. We associate with boundary value problem (1) a bounded linear operatorand we define for the family of limit operators....

We consider underwater acoustic fields generated by non-uniformly moving modulated sources. We introduce a large parameter which characterizing simultaneously a slowness of variations of source amplitudes, and their horizontal velocities, and a large distances between sources and receivers with respect to the characteristic length of the acoustic w...

The main aim of the paper is to study the Fredholm property, essential spectrum, and invertibility of some operators of the Mathematical Physics, such that the Schrödinger and Dirac operators with complex elec- tric potentials, and Maxwell operators in absorbing at infinity media. This investigation is based on the limit operators method, and the u...

The paper is devoted to the of Fredholm property of pseudodifferential operators acting in the spaces of Bessel potentials connected with variable exponent Lebesgue spaces on smooth compact manifolds and non compact manifolds with conical structure at infinity.

We consider acoustic diffraction by graphs Γ embedded in ℝ2 and periodic with respect to an action of the group ℤ
n
, n = 1, 2. The diffraction problem is described by the Helmholtz equation with variable nonperiodic bounded coefficients and nonperiodic transmission conditions on the graph Γ. We introduce single and double layer potentials on Γ gen...

We apply the complex two-dimensional stationary-phase method for problems of electromagnetic waves radiation from the moving modulated sources in dispersive and lossy medium. The explicit formulas for fields generated by a moving source, the Doppler effect, and the retarded time are obtained. We apply the developed approach to numerical study of th...

The spectral parameter power series method is applied to problems of underwater sound propagation in the stratified ocean. This method is based on representations of solutions of vertical Sturm–Liouville problems of the form of power series with respect to the spectral parameter, and it gives an effective dispersion equation for wave numbers and ex...

Let be a smooth unbounded domain in conical at infinity, We consider general transmission problems defined by a differential equation(1 )and transmission conditions on the boundary (2 )where the coefficients are discontinuous on functions, such that the space of infinitely differentiable functions in bounded with all derivatives, is a jump of the f...

In the present work, we analyze the electromagnetic field generated by a modulated point source in a planarly layered waveguide, in the far field region. On the basis of the two-dimensional stationary phase method, we obtain expressions for the asymptotics of the field at large distance from the source and a large value of the time. The analysis re...

This work addresses the analysis of an isotropic planarly layered waveguide consisting of an inhomogeneous core that is enclosed between two homogeneous layers forming the cladding. The analysis relies on an auxiliary one-dimensional spectral problem that is intimately linked with the scalar wave equation for planarly layered media. We construct th...

The paper is devoted to the diffraction of graphs [Inline formula] imbedded in [Inline formula] periodic with respect to the action of the group [Inline formula] We consider the Helmholtz equation (1) where, [Inline formula] is the frequency of a medium, [Inline formula] is the frequency of harmonic vibrations, and [Inline formula] is the velocity...

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...

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.

The paper is devoted to applications the quaternionic analysis and the
two-dimensional stationary phase method for problems of electromagnetic
waves propagation from moving modulated sources in dispersive chiral
media. This paper is based in the article [2].

We consider a diffraction problem in a multi-connected domain ℜ2 \ Γ, where Γ is an oriented graph with finitely many edges some of which are infinite. The problem is described by the Helmholtz equation (1) where ρ and k are functions bounded together with all derivatives, and by the transmission conditions (2)(3) where V is the set of vertices, a...

The main aim of the paper is the study of essential spectra of electromagnetic Schrödinger operators with variable potentials in cylindric domains , where is a bounded domain with a smooth boundary provided by admissible boundary conditions. Applying the limit operators method, we obtain explicit estimates of the essential spectrum for a wide class...

A representation for the fields generated by moving sources in chiral media in the form of double time-frequency oscillating integrals is obtained by using quaternionic analysis methods. Some additional assumptions concerning the source allow us to introduce a large dimensionless parameter λ > 0 which characterizes simultaneously the slowness of va...

The radiation of a nanosource placed in a coated microsphere with conventional and metamaterial layers having a negative refraction index (NIM) is studied. We consider also that a NIM defect is embedded in such a structure. Our calculations show strong enhancement of the optical field strength assisted by NIM defect. In a resonant case the optical...

We consider double-layer potential type operators acting in weighted variable exponent Lebesgue space \( L^{p(.)}(\Gamma,w)\) on some composed curves with oscillating singularities. We obtain a Fredholm criterion for operators \( A=aI+bD_{g.\Gamma}:L^{p(.)}(\Gamma,w)\rightarrow L^{p(.)}(\Gamma,w) \; {\rm where}{D_{g,\Gamma}} \) is the operator of t...

The time-frequency integrals and the two-dimensional stationary phase method
are applied to study the electromagnetic waves radiated by moving modulated
sources in dispersive media. We show that such unified approach leads to
explicit expressions for the field amplitudes and simple relations for the
field eigenfrequencies and the retardation time t...

The paper is devoted to singular integral operators acting on weighted variable exponent Lebesgue spaces on certain composed Carleson curves. Necessary and sufficient conditions for Fredholmness and an index formula are obtained.

The propagation of electromagnetic waves issued by modulated moving sources of the form \(j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right)\) is considered, where j(t, x) stands for the current density vector, x = (x
1, x
2, x
3) ∈ ℝ3 for the space variables, t ∈ ℝ...

The Fredholm property and the essential spectrum of pseudo-differential operators (PDO) with operator-valued symbols is studied. The method of limit operators is developed and the essential spectrum of operators of quantum waveguides is considered. Two types of electric potentials are analyzed, slowly oscillating at infinity and perturbations of pe...

The main aim of the paper is Fredholm properties of a class of bounded linear
operators acting on weighted Lebesgue spaces on an infinite metric graph
$\Gamma$ which is periodic with respect to the action of the group ${\mathbb
Z}^n$. The operators under consideration are distinguished by their local
behavior: they act as (Fourier) pseudodifferenti...

We derive a dispersion equation for determining eigenvalues of inhomogeneous quantum wells in terms of spectral parameter power series and apply it for the numerical treatment of eigenvalue problems. The method is algorithmically simple and can be easily implemented using available routines of such environments for scientific computing as MATLAB.

We consider the class of h-pseudo-differential operators $$\begin{array}{lcr} \text{A}_{h} u(x) = Op_h (a)u(x) \\
\quad\quad\quad{ = (2}\pi {h)}^{{ - n}} \int\limits_{\mathbb{R}^{2n}
} {a(x,\xi,h)} u(y)e^{\frac{i}{h}(x - y).\xi } dyd\xi,u \in
S(\mathbb{R}^n,\mathcal{H}_1 )\end{array},$$ where the symbol a has values in the space of bounded linear o...

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L
p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the sp...

The paper is devoted to exponential estimates of eigenfunctions of the discrete spectrum of matrix Schrödinger operators with
variable potentials, and Dirac operators for nonhomogeneous media with variable light speed and variable electric and magnetic
potentials, For the study of exponential estimates we apply methods developed in our recent paper...

This article is devoted to the problems of invertibility and Fredholmness of operators in the weighted Wiener algebra acting in the weighted spaces of vector-functions on the lattice (μℤ) = {x∈ℝ : x = μy,y∈ℤ} for small μ>0. We consider the discrete Schrödinger operators on (μℤ) as applications and study the relation between the spectra of Schröding...

We consider the invertibility of parabolic pseudodifferential operators in
exponential weighted\ Sobolev spaces. We suppose that the symbol $a$ of \ the
operator $Op(a)$ is analytically extended with respect to the impulse variable
in an unbounded tube domain $\mathbb{R}^{n}+iD$ and satisfies conditions of
uniform parabolicity . We prove that under...

We consider a class of pseudodifferential operators with operator-valued symbols a(x,ξ) under the assumption that a(x,ξ) can be analytically extended with respect to ξ onto a tube domain ℝ n +iℬ where ℬ is a convex bounded domain in ℝ n containing the origin. The main result of the paper is exponential estimates at infinity of solutions of pseudodi...

We consider the Fredholm property and essential spectrum of pseudodifferential operators with symbols having values in the space of bounded operators acting from a separable Hilbert space ${\mathcal{H}}_{1}$ into a separable Hilbert space ${\mathcal{H}}_{2}$ and satisfying additional estimates. There are many problems in mathematical physics which...

We studied the radial distribution of the electromagnetic field and the spectrum of the eigenfrequencies in coated microspheres with alternating quasiperiodic layers having left-handed (LH) materials included. At an increase of the quasiperiodicity parameter $\gamma$, the boundaries of the spectral band gaps acquire the intended shape. When $\gamma...

Non-closed algebras ${\mathfrak{A}}_{\delta,\gamma}$ of pseudodifferential operators with slowly oscillating Lipschitz symbols in $\Lambda^{SO}_{\delta,\gamma}({\mathbb R}\times{\mathbb R})$ with $\delta,\gamma\in(1/2,1]$ and the minimal $C^*$-algebra ${\mathfrak{A}}$ containing all ${\mathfrak{A}}_{\delta,\gamma}$ are studied on the Lebesgue space...

The main aim of the paper is the application of the calculus of pseudodifferential operators on the lattice
(h \mathbbZ)n={ x Î \mathbbRn:x=hy,y Î \mathbbZn}{(h \mathbb{Z})^{n}=\{ x\in \mathbb{R}^{n}:x=hy,y\in\mathbb{Z}^{n}\}} depending on a small parameter h>0 to exponential estimates of solutions of difference equations on
( h\mathbbZ)n{( h\m...

Let $\Gamma$ be a finitely generated discrete exact group. We consider operators on $l^2(\Gamma)$ which are composed by operators of multiplication by a function in $l^\infty (\Gamma)$ and by the operators of left-shift by elements of $\Gamma$. These operators generate a $C^*$-subalgebra of $L(l^2(\Gamma))$ the elements of which we call band-domina...

The main aim of the paper is the investigation of a relation between the essential spectrum and the exponential decay at infinity
of eigenfunctions of the lattice analogs of Schrödinger and Dirac operators.

Let be an open set, and where We consider pseudodifferential operators in domain Ω+ with double symbols which have singularities near and super exponential growths at infinity. We suppose that symbols have analytic extension with respect to the variable dual to the time in the lower complex half-plane. We construct the theory of invertibility of su...

We consider the class of matrix h-pseudodifferential operators O(Ph) (a) with symbols a = (a(ij)) (N)(i,j=1) , where the coefficients a(ij) is an element of C(infinity)(R(x)(n) X R(xi)(n)) circle times C(0, 1] satisfy the estimates vertical bar partial derivative(beta)(chi)partial derivative(alpha)(xi)a(ij)(chi, xi, h)vertical bar <= C(alpha beta)x...

The paper deals with the study of Fredholm property and essential spectrum of general difference (or band) operators acting
on the spaces l
p
(X) on a discrete metric space X periodic with respect to the action of a finitely generated discrete group. The Schrödinger operator on a combinatorial periodic
graph is a prominent example of a band operat...

This paper is devoted to estimates of the exponential decay of eigenfunctions
of difference operators on the lattice Z^n which are discrete analogs of the
Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our
investigation of the essential spectra and the exponential decay of
eigenfunctions of the discrete spectra is based on the calcu...

The basic goal of this paper is to derive exponential estimates for eigenfunctions of matrix elliptic differen- tial operators onn . We show that these estimates are related to the essential spectra of differential operators from some family depending on the original operator and an exponential weight describing the behavior of the eigenfunctions a...

The paper is devoted to the investigation of the Helmholtz operators
describing the propagation of acoustic waves in non-homogeneous space. We consider the operator A with a wave number k such that
where k0 is a positive function, k± are complex constants with ℑ(k)>0. The Helmholtz operator A with such wave number describes the propagation of aco...

## Projects

Projects (2)

Computation of propagated modes.
Design of dispersion controlled fibers.
Analysis of pulse modulation.