Minimal Symmetric Operator Research Articles

On extensions of matrix-valued Hahn–Sturm–Liouville operators

In this paper, we study matrix-valued Hahn–Sturm–Liouville equations. We give an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system, and some properties of these operators are investigated. Finally, we characterize extensions (maximal dissipative, maximal accumulative and self-adjoint) of the minimal symmetric operator.

Dilation, model, scattering and spectral problems of second-order matrix difference operator

In the Hilbert space ?2 ?(Z; E) (Z := {0,? 1, ? 2, ...}, dim E = N < ?), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at ? ?) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functional model of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved.

Conformable fractional Sturm–Liouville problems on time scales

We consider a conformable fractional Sturm–Liouville problem on bounded time scales. Firstly, the existence and uniqueness of solutions of the conformable fractional Sturm–Liouville problem are proved. Maximal and minimal symmetric operators are introduced and some results are given. Finally, the Green's function of this problem is constructed, and an eigenfunction expansion is given.

Extensions of the matrix-valued q−Sturm–Liouville operators

In this paper, we investigate the matrix-valued$\ q-$Sturm--Liouville problems. We establish an existence and uniqueness result. Later, we introduce the corresponding maximal and minimal operators for this system. Moreover, we give a criterion under which these operators are self-adjoint. Finally, we characterize extensions (maximal dissipative, maximal accumulative, and self-adjoint) of the minimal symmetric operator.

On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

We study realizations generated by the original Weyl–Titchmarsh functions $$m_\infty (z)$$ and $$m_\alpha (z)$$ . It is shown that the Herglotz–Nevanlinna functions $$(-\,m_\infty (z))$$ and $$(1/m_\infty (z))$$ can be realized as the impedance functions of the corresponding Shrodinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions $$(-\,m_\alpha (z))$$ and $$(1/m_\alpha (z))$$ . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

Fourth order differential operators with distributional potentials

In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.

Some singular third‐order boundary value problems

In this paper, we consider some singular formally symmetric (self‐adjoint) boundary value problems generated by a singular third‐order differential expression and separated and coupled boundary conditions. In particular, we consider that the minimal symmetric operator generated by the third‐order differential expression has the deficiency indices (3,3). We investigate same spectral properties related with these problems, and we introduce a method to find the resolvent operator.

Functional Model and Spectral Analysis of Discrete Singular Hamiltonian System

A space of boundary values is constructed for a minimal symmetric operator, generated by a discrete singular Hamiltonian system, acting in the Hilbert space $\ell_{\mathbf{A}}^{2}(\mathbb{N}_{0}; E \oplus E)$ ($\mathbb{N}_{0} = \{ 0,1,2,\ldots \}$, $\dim E = m \lt \infty$) with maximal deficiency indices $(m,m)$ (in limit-circle case). A description of all maximal dissipative, maximal accumulative, self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a self-adjoint dilation of a maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We establish a functional model of the dissipative operator and construct its characteristic function in terms of the scattering matrix of the dilation. Finally, we prove the theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the dissipative operator.

On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients

Asymptotic formulas as x→∞ are obtained for a fundamental system of solutions to equations of the form $$l\left( y \right): = {\left( { - 1} \right)^n}{\left( {p\left( x \right){y^{\left( n \right)}}} \right)^{\left( n \right)}} + q\left( x \right)y = \lambda y,x \in [1,\infty )$$ , where p is a locally integrable function representable as $$p\left( x \right) = {\left( {1 + r\left( x \right)} \right)^{ - 1}},r \in {L^1}\left( {1,\infty } \right)$$ , and q is a distribution such that q = σ(k) for a fixed integer k, 0 ≤ k ≤ n, and a function σ satisfying the conditions $$\sigma \in {L^1}\left( {1,\infty } \right)ifk < n,$$ $$\left| \sigma \right|\left( {1 + \left| r \right|} \right)\left( {1 + \left| \sigma \right|} \right) \in {L^1}\left( {1,\infty } \right)ifk = n$$ . Similar results are obtained for functions representable as $$p\left( x \right) = {x^{2n + v}}{\left( {1 + r\left( x \right)} \right)^{ - 1}},q = {\sigma ^{\left( k \right)}},\sigma \left( x \right) = {x^{k + v}}\left( {\beta + s\left( x \right)} \right)$$ , for fixed k, 0 ≤ k ≤ n, where the functions r and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression l(y) (for real functions p and q) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case n = 1.

Spectral problems of non-self-adjoint q-Sturm-Liouville operators in limit-point case

In this study, dissipative singular q-Sturm-Liouville operators are studied in the Hilbert space $\mathscr{L}_{r,q}^{2}$(Rq,+), that the extensions of a minimal symmetric operator in limit-point case. We construct a self-adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax-Phillips. Then, we create a functional model of the maximal dissipative operator via the incoming spectral representation and define its characteristic function in terms of the Weyl-Titchmarsh function (or scattering function of the dilation) of a self-adjoint q-Sturm-Liouville operator. Finally, we prove the theorem on completeness of the system of eigenfunctions and associated functions (or root functions) of the dissipative q-Sturm-Liouville operator.

Asymptotics of Solutions of One Class of Linear Differential Equations of Arbitrary Order

In this paper, we study the problem on the asymptotic behavior of solutions of the equation l n y = λy of arbitrary order (even or odd) with complex-valued coefficients in a neighborhood of the point +∞. The result obtained allows one to find defect numbers of the corresponding closed minimal symmetric differential operator.

Extensions of symmetric infinite Jacobi operator

A space of positive boundary values is constructed for a positive definite minimal symmetric operator generated by an infinite Jacobi matrix in limit-circle case. A description of all solvable, maximal dissipative (accumulative) solvable, self-adjoint solvable, positive definite self-adjoint and non-positive definite self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity.

NONSELFADJOINT SINGULAR STURM-LIOUVILLE OPERATORS IN LIMIT-CIRCLE CASE

In this paper, we study the maximal dissipative singular Sturm-Liouville operators (in Weyl's limit-circle case at singular point $b$) acting in the Hilbert space $L_{w}^{2}\left[ a,b\right)$ ($-\infty \lt a\lt b\leq \infty$ ). In fact, we consider all extensions of a minimal symmetric operator and we investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at $a$' and 'dissipative at $b$'. In both cases, we construct a selfadjoint dilation of the maximal dissipative operator and determine its incoming and outgoing spectral representations. This representations make it possible to determine the scattering matrix ofthe dilation in terms of the Titchmarsh-Weyl function of a selfadjoint Sturm-Liouville operator. We also construct a functional model of the maximal dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Titchmarsh-Weyl function). Finally we prove theorems on the completeness of the eigenfunctions and associated functions of the maximal dissipative Sturm-Liouville operators.

Nonself‐Adjoint Second‐Order Difference Operators in Limit‐Circle Cases

We consider the maximal dissipative second‐order difference (or discrete Sturm‐Liouville) operators acting in the Hilbert space (ℤ: = {0, ±1, ±2, …}), that is, the extensions of a minimal symmetric operator with defect index (2,2) (in the Weyl‐Hamburger limit‐circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at −∞” and “dissipative at ∞.” In each case, we construct a self‐adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh‐Weyl function of the self‐adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

Deficiency indices and spectrum of self-adjoint extensions of some classes of differential operators

Problems relating to the asymptotic behaviour in the neighbourhood of the point and in the neighbourhood of the origin of a solution of an equation of arbitrary (even or odd) order with complex-valued coefficients are studied. It is assumed here that the coefficients of the quasidifferential expression have the following property: if one reduces the equation to a system of first-order differential equations, then one can transform that system to a system of differential equations with regular singular point at or . The results obtained allow one to determine the deficiency indices of the corresponding minimal symmetric differential operators and the structure of the spectrum of self-adjoint extensions of these operators. In addition, on the basis of refined asymptotic formulae for solutions to the equation the deficiency numbers of a certain differential operator generated by a differential expression with leading coefficient vanishing in the interior of the interval in question are found.

Extensions, Dilations and Functional Models of Infinite Jacobi Matrix

A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative operators.

Dissipative Sturm?Liouville Operators in Limit-Point Case

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space Lw2[a,b) (−∞<a<b≤∞), that the extensions of a minimal symmetric operator in Weyl’s limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators.

Dissipative discrete Hamiltonian systems

A space of boundary values is constructed for minimal symmetric operator, generated by discrete Hamiltonian system, acting in the Hilbert space l 2 A (ℤ E ⊕ E) (ℤ={0, ±1, ±2, …}, dim E = n < ∞) with deficiency indices ( n, n) (in limit-circle case at ±∞ and limit point case at ∓∞). A description of all maximal dissipative, maximal accretive, and self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at ∞’. In each of these cases we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation in terms of the Titchmarsh-Weyl matrix-valued function of the self-adjoint operator. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of dilation. Finally, we prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

Extensions, Dilations and Functional Models of Dirac Operators

A space of boundary values is constructed for minimal symmetric Dirac operator in the Hilbert space \(L_A^2 (( - \infty ,\infty );\mathbb{C}^2 )\) with defect index (2,2) (in Weyl’s limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with separated boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at +∞’. In each of these cases we construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix. We construct a functional model of the maximal dissipative operator and define its characteristic function. We prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

Correct selfadjoint and positive extensions ofnondensely defined minimal symmetric operators

Let A0 be a closed, minimal symmetric operator from a Hilbert space ℍ into ℍ with domain not dense in ℍ. Let also be a correct selfadjoint extension of A0. The purpose of this paper is (1) to characterize, with the help of , all the correct selfadjoint extensions B of A0 with domain equal to , (2) to give the solution of their corresponding problems, (3) to find sufficient conditions for B to be positive (definite) when is positive (definite).