arXiv subjects

Vadim Mogilevskii

Publications and source records attributed to Vadim Mogilevskii.

15 recordsLinked to original sources

arXiv2020

On eigenfunction expansions of differential equations with degenerating weight

Let $A$ be a symmetric operator. By using the method of boundary triplets we parameterize in terms of a Nevanlinna parameter $τ$ all exit space extensions $\wt A=\wt A^*$ of $A$ with the discrete spectrum $\s(\wt A)$ and characterize the Shtraus family of $\wt A$ in terms of abstract boundary conditions. Next we apply these results to the eigenvalue problem for the $2r$-th order differential equation $l[y]= ł\D(x)y$ on an interval $[a,b), \; -\infty <a<b\leq \infty,$ subject to $ł$-depending separated boundary conditions with entire operator-functions $C_0(ł) $ and $C_1(ł)$, which form a Nevanlinna pair $(C_0,C_1)$. The weight $\D(x)$ is nonnegative and may vanish on some intervals $(\a,\b)\subset \cI$. We show that in the case when the minimal operator of the equation has the discrete spectrum (in particular, in the case of the quasiregular equation) the set of eigenvalues of the eigenvalue problem is an infinite subset of $\bR$ without finite limit points and each function $y\in\LI$ admits the eigenfunction expansion $y(x)=\sum_{k=1}^\infty y_k (x)$ converging in $\LI$. Moreover, we give an explicit method for calculation of eigenfunctions $y_k$ in this expansion and specify boundary conditions on $y$ implying the uniform convergence of the eigenfunction expansion of y. These results develop the known ones obtained for the case of the positive weight $\D$ and the more restrictive class of $ł$-depending boundary conditions.

math.FA↗

arXiv2020

On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

We study pseudospectral and spectral functions for Hamiltonian system $Jy'-B(t)=λΔ(t)y$ and differential equation $l[y]=λΔ(t)y$ with matrix-valued coefficients defined on an interval $\mathcal{I}=[a,b)$ with the regular endpoint $a$. It is not assumed that the matrix weight $Δ(t)\geq 0$ is invertible a.e. on $\mathcal{I}$. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization $σ=σ_τ$ of all pseudospectral and spectral functions $σ$ by means of a Nevanlinna parameter $τ$ and single out in terms of $τ$ and boundary conditions the class of functions $y$ for which the inverse Fourier transform $y(t)=\int\limits_{\mathbb{R}} φ(t,s)\, dσ(s) \widehat y(s)$ converges uniformly. We also show that for scalar equation $l[y]=λΔ(t)y$ the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm - Liouville equation $-(p(t)y')'+q(t)y=λΔ(t) y$ to such equations with arbitrary coefficients $p(t)$ and $q(t)$ and arbitrary non trivial weight $Δ(t)\geq 0$.

math.FA↗

arXiv2018

On compressions of self-adjoint extensions of a symmetric linear relation

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with equal deficiency indices $n_\pm (A)\leq\infty$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an exit space extension of $A$; such an extension is called finite-codimensional if $\dim (\wt\gH\ominus\gH)< \infty$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of exit space extensions $\wt A=\wt A^*$. For a certain class of extensions $\wt A$ we parameterize the compressions $C (\wt A)$ by means of abstract boundary conditions. This enables us to characterize various properties of $C (\wt A)$ (in particular, self-adjointness) in terms of the parameter for $\wt A$ in the Krein formula for resolvents. We describe also the compressions of a certain class of finite-codimensional extensions. These results develop the results by A. Dijksma and H. Langer obtained for a densely defined symmetric operator $A$ with finite deficiency indices.

math.FA↗

arXiv2018

Weyl Solutions and J-selfadjointness for Dirac operators

We consider a non-selfadjoint Dirac-type differential expression \begin{equation} D(Q)y:= J_n \frac{dy}{dx} + Q(x)y, \quad\quad\quad (1) \end{equation} with a non-selfadjoint potential matrix $Q \in L^1_{loc}({\mathcal I},\mathbb{C}^{n\times n})$ and a signature matrix $J_n =-J_n^{-1} = -J_n^*\in \mathbb{C}^{n\times n}$. Here ${\mathcal I}$ denotes either the line $\mathbb{R}$ or the half-line $\mathbb{R}_+$. With this differential expression one associates in $L^2(\mathcal I,\mathbb{C}^{n})$ the (closed) maximal and minimal operators $D_{\max}(Q)$ and $D_{\min}(Q)$, respectively. One of our main results states that $D_{\max}(Q) = D_{\min}(Q)$ in $L^2(\mathbb{R},\mathbb{C}^{n})$. Moreover, we show that if the minimal operator $D_{\min}(Q)$ in $L^2(\mathbb{R},\mathbb{C}^{n})$ is $j$-symmetric with respect to an appropriate involution $j$, then it is $j$-selfadjoint. Similar results are valid in the case of the semiaxis $\mathbb{R}_+$. In particular, we show that if $n=2p$ and the minimal operator $D_{\min}(Q)$ in $L^2(\mathbb{R}_+,\mathbb{C}^{2p})$ is $j$-symmetric, then there exists a $2p\times p$-Weyl-type matrix solution $Ψ(z, \cdot)\in L^2(\mathbb{R}_+,\mathbb{C}^{2p\times p})$ of the equation $D^+_{\max}(Q)Ψ(z, \cdot)= zΨ(z, \cdot)$. A similar result is valid for the expression (1) with a potential matrix having a bounded imaginary part. This leads to the existence of a unique Weyl function for the expression (1). The differential expression (1) is of significance as it appears in the Lax formulation of the vector-valued nonlinear Schr{ö}dinger equation.

math.SP↗

arXiv2016

Spectral and pseudospectral functions of various dimensions for symmetric systems

The main object of the paper is a symmetric system $J y'-B(t)y=ł\D(t) y$ defined on an interval $\cI=[a,b) $ with the regular endpoint $a$. Let $\f(\cd,ł)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=\int\limits_\cI \f^*(t,s)\D(t)f(t)\,dt$ be the Fourier transform of the function $f(\cd)\in L_\D^2(\cI)$. We define a pseudospectral function of the system as a matrix-valued distribution function $\s(\cd)$ of the dimension $n_\s$ such that $V$ is a partial isometry from $L_\D^2(\cI)$ to $L^2(\s;\bC^{n_\s})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_\s$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_\s$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.

math.FA↗

arXiv2014

On spectral and pseudospectral functions of first-order symmetric systems

We consider general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$. A distribution matrix-valued function $\Si(s), \; s\in\bR,$ is called a spectral (pseudospectral) function of such a system if the corresponding Fourier transform is an isometry (resp. partial isometry) from $\LI$ into $L^2(\Si)$. The main result is a parametrization of all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter $τ$. Similar parameterizations for various classes of boundary problems have earlier been obtained by Kac and Krein, Fulton, Langer and Textorius, Sakhnovich and others.

math.FA↗

arXiv2014

On generalized resolvents and characteristic matrices of first-order symmetric systems

We study general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$ and singular endpoint $b$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the corresponding minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. We describe all generalized resolvents $y=R(ł)f, \; f\in\LI,$ of $\Tmi$ in terms of boundary problems with $ł$-depending boundary conditions imposed on regular and singular boundary values of a function $y$ at the endpoints $a$ and $b$ respectively. We also parametrize all characteristic matrices $\Om(ł)$ of the system immediately in terms of boundary conditions. Such a parametrization is given both by the block representation of $\Om(ł)$ and by the formula similar to the well-known Krein formula for resolvents. These results develop the uStraus' results on generalized resolvents and characteristic matrices of differential operators.

math.FA↗

arXiv2013

On eigenfunction expansions of first-order symmetric systems and ordinary differential operators of an odd order

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b\rangle $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ satisfy $n_+(\Tmi)< n_-(\Tmi)$. We define $ł$-depending boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate an exit space self-adjoint extension $\wt T$ of $\Tmi$ and the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. By using $m$-function we obtain the eigenfunction expansion with the spectral function $\Si(\cd)$ of the minimally possible dimension and characterize the case when spectrum of $\wt T$ is defined by $\Si(\cd)$. Moreover, we parametrize all spectral functions in terms of a Nevanlinna type boundary parameter. Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh-Weyl theory of such operators.

math.FA↗

arXiv2013

On Titchmarsh-Weyl functions and eigenfunction expansions of first-order symmetric systems

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $\cI=[a,b> $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. By using a Nevanlinna boundary parameter $τ=τ(ł)$ at the singular endpoint $b$ we define self-adjoint and $ł$-depending Nevanlinna boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. By using $m$-function we obtain the Fourier transform $V:\LI\to L^2(\Si)$ with the spectral function $\Si(\cd)$ of the minimally possible dimension. If $V$ is an isometry, then the (exit space) self-adjoint extension $\wt T$ of $\Tmi$ induced by the boundary problem is unitarily equivalent to the multiplication operator in $L^2(\Si)$; hence the spectrum of $\wt T$ is defined by the spectral function $\Si(\cd)$. We show that all the objects of the boundary problem are determined by the parameter $τ$, which enables us to parametrize all spectral function $\Si(\cd) $ immediately in terms of $τ$. Similar results for various classes of boundary problems were obtained by Kac and Krein, Fulton, Hinton and Shaw and other authors.

math.FA↗

arXiv2012

Unitary equivalence of proper extensions of a symmetric operator and the Weyl function

Let $A$ be a densely defined simple symmetric operator in $\gH$, let $Π=\bt$ be a boundary triplet for $A^*$ and let $M(\cd)$ be the corresponding Weyl function. It is known that the Weyl function $M(\cd)$ determines the boundary triplet $Π$, in particular, the pair ${A,A_0}$, where $A_0:= A^*\lceil\ker\G_0 (= A^*_0)$, uniquely up to unitary similarity. At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to weak similarity. In this paper we consider symmetric dual pairs ${A,A}$ generated by $A\subset A^*$ and special boundary triplets $\wtΠ$ for ${A,A}$. We are interested whether the result on unitary similarity remains valid provided that the Weyl function corresponding to $\wtΠ$ is $\wt M(z)= K^*(B-M(z))^{-1} K,$ where $B$ is some non-self-adjoint bounded operator in $\cH$. We specify some conditions in terms of the operators $A_0$ and $A_B= A^*\lceil \ker(\G_1-B\G_0)$, which determine uniquely (up to unitary equivalence) the pair ${A,A_B}$ by the Weyl function $\wt M(\cd)$. Moreover, it is shown that under some additional assumptions the Weyl function $M_Π(\cdot)$ of the boundary triplet $Π$ for the dual pair $\DA$ determines the triplet $Π$ uniquely up to unitary similarity. We obtain also some negative results demonstrating that in general the Weyl function $\wt M(\cd)$ does not determine the operator $A_B$ even up to similarity.

math.FA↗

arXiv2012

On Titchmarsh-Weyl functions of first-order symmetric systems with arbitrary deficiency indices

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $[a,b> $ with the regular endpoint $a$. The deficiency indices $n_\pm$ of the corresponding minimal relation $\Tmi$ may be arbitrary (possibly unequal). Our approach is based on the concept of a decomposing boundary triplet, which enables one to parametrize various classes of extensions of $\Tmi$ (self-adjoint, $m$-dissipative, etc.) in terms of boundary conditions imposed on regular and singular values of a function $y\in \dom \tma$ at the endpoints $a$ and $b$ respectively. In particular, we describe self-adjoint and $ł$-depending Nevanlinna boundary conditions which are analogs of separated ones for Hamiltonian systems. With a boundary value problem involving such conditions we associate the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. In the simplest case of minimal (unequal) deficiency indices $n_\pm$ the $m$-function $m(\cd)$ coincides with the rectangular Titchmarsh-Weyl coefficient introduced by Hinton and Schneider. We parametrize all $m$-functions in terms of the Nevanlinna boundary parameter at the endpoint $b$ by means of the formula similar to the known Krein formula for resolvents. Application of these results to differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh-Weyl theory of such operators.

math.FA↗

arXiv2011

Boundary relations and boundary conditions for general (not necessarily definite) canonical systems with possibly unequal deficiency indices

We investigate in the paper general (not necessarily definite) canonical systems of differential equation in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary relation $\G:\gH^2\to\HH$ for $A^*$, where $\gH$ is a Hilbert space, $A$ is a symmetric linear relation in $\gH, \cH_0$ is a boundary Hilbert space and $\cH_1$ is a subspace in $\cH_0$. Unlike known concept of a boundary relation (boundary triplet) for $A^*$ our definition of $\G$ is applicable to relations $A$ with possibly unequal deficiency indices $n_\pm(A)$. Next we develop the known results on minimal and maximal relations induced by the general canonical system $ J y'(t)-B(t)y(t)=\D (t)f(t)$ on an interval $\cI=(a,b),\; -\infty\leq a<b\leq\infty $ and then by using a special (so called decomposing) boundary relation for $\Tma$ we describe in terms of boundary conditions proper extensions of $\Tmi$ in the case of the regular endpoint $a$ and arbitrary (possibly unequal) deficiency indices $n_\pm (\Tmi)$. If the system is definite, then decomposing boundary relation $\G$ turns into the decomposing boundary triplet $Π=\bt$ for $\Tma$. Using such a triplet we show that self-adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, we describe all such conditions in the compact form. These results are generalizations of the known results by Rofe-Beketov on regular differential operators. We characterize also all maximal dissipative and accumulative separated boundary conditions, which exist for arbitrary (not necessarily Hamiltonian) definite canonical systems.

math.FA↗

arXiv2010

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let $\gH$ be a Hilbert space and let $A$ be a simple symmetric operator in $\gH$ with equal deficiency indices $d:=n_\pm(A)<\infty$. We show that if, for all $ł$ in an open interval $I\subset\bR$, the dimension of defect subspaces $\gN_ł(A)(=\Ker (A^*-ł))$ coincides with $d$, then every self-adjoint extension $\wt A\supset A$ has no continuous spectrum in $I$ and the point spectrum of $\wt A$ is nowhere dense in $I$. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator. Moreover, we show in the paper, that an old conjecture by Hartman and Wintner on the spectrum of a self-adjoint Sturm - Liouville operator is not valid.

math.FA↗

arXiv2010

Minimal spectral functions of an ordinary differential operator

Let $l[y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b> \;(b\leq \infty)$ and let $L_0$ be the corresponding minimal operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation $l[y]-ły=f\;(f\in L_2[0,b>)$ and the Nevanlinna $ł$-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the $m$-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e., all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices $n_\pm(L_0)$ and not decomposing boundary conditions) the known estimate of the spectral multiplicity of the (exit space) selfadjoint extension $\wt A\supset L_0$. The results of the paper are obtained for expressions $l[y]$ with operator valued coefficients and arbitrary (equal or unequal) deficiency indices $n_\pm(L_0)$.

math.FA↗

arXiv2009

On generalized resolvents and characteristic matrices of differential operators

The main objects of our considerations are differential operators generated by a formally selfadjoint differential expression of an even order on the interval $[0,b> (b\leq \infty)$ with operator valued coefficients. We complement and develop the known Shtraus' results on generalized resolvents and characteristic matrices of the minimal operator $L_0$. Our approach is based on the concept of a decomposing boundary triplet which enables to establish a connection between the Straus' method and boundary value problems (for singular differential operators) with a spectral parameter in a boundary condition. In particular we provide a parametrization of all characteristic matrices $\Om (ł)$ of the operator $L_0$ immediately in terms of the Nevanlinna boundary parameter $τ(ł)$. Such a parametrization is given in the form of the block-matrix representation of $\Om (ł)$ as well as by means of the formula for $\Om (ł)$ similar to the well known Krein-Naimark formula for generalized resolvents. In passing we obtain the representation of canonical and generalized resolvents in the form of integral operators with the operator kernel (the Green function) defined in terms of fundamental operator solutions of the equation $l[y]-ły=0$.

math.FA↗