Limit-circle Case Research Articles

A growth estimate for the monodromy matrix of a canonical system

We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein–de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum. We prove a general and flexible upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work by R. Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.

Nonlinear singular Hahn-Sturm-Liouville problems on [ω_0, ∞)

In this paper, we investigate a nonlinear Hahn-Sturm-Liouville problem on [ω0, ∞) under the assumption that the Hahn-Sturm-Liouville equation is in the limit-circle case at ∞. The existence and uniqueness of the solutions for these problems are obtained.

Self-adjoint and non-self-adjoint extensions of symmetric q-Sturm-Liouville operators

Aspace of boundary values is constructed for minimal symmetric regular and singular q-Sturm- Liouville operators in limit-point and limit-circle cases. A description of all maximal dissipative, maximal accumulative, self-adjoint, and other extensions of such symmetric q-Sturm-Liouville operators is given in terms of boundary conditions.

Research on singular Sturm–Liouville spectral problems with a weighted function

As early as 1910, Weyl gave a classification of the singular Sturm–Liouville equation, and divided it into the Limit Point Case and the Limit Circle Case at infinity. This led to the study of singular Sturm–Liouville spectrum theory. With the development of applications, the importance of singular Sturm–Liouville problems with a weighted function becomes more and more significant. This paper focuses on the study of singular Sturm–Liouville problems with a weighted function. Finally, an example of singular Sturm–Liouville problems with a weighted function is given.

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.

Spectral problems of dissipative singular q-Sturm-Liouville operators in limit-circle case

We consider the dissipative singular q-Sturm-Liouville operators acting in the Hilbert space L2 w,q(R+), that the extensions of a minimal symmetric operator with deficiency indices (2, 2) (in limit-circle case).We construct a self-adjoint dilation of the 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 Weyl-Titchmarsh function of a self-adjoint q-Sturm-Liouville operator. We also construct a functional model of the dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl-Titchmarsh function). Theorems on the completeness of the system of or root functions of the dissipative and accumulative q-Sturm-Liouville operators are proved.

INTERVAL-TYPE CRITERIA OF LIMIT-POINT CASE FOR CONFORMABLE FRACTIONAL STURM-LIOUVILLE OPERATORS

In this paper, the classification of limit-point case and limit-circle case for the 2<i>α</i>-order conformable fractional Sturm-Liouville operator<br/>${\ell _\alpha }(y) = - {T_\alpha }(p{T_\alpha }y) + qy,x \in [a,\infty ),a > 0$<br/>is considered. Two interval criteria of limit-point case in the frame of conformable fractional derivatives are obtained. Examples illustrating the main results are presented.

Titchmarsh-Weyl Theory of the Singular Hahn-Sturm-Liouville Equation

In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

On a Class of Integral Systems

We study spectral problems for two-dimensional integral system with two given non-decreasing functions R, W on an interval [0, b) which is a generalization of the Krein string. Associated to this system are the maximal linear relation T_{max } and the minimal linear relation T_{min } in the space L^2(dW) which are connected by T_{max }=T_{min }^*. It is shown that the limit point condition at b for this system is equivalent to the strong limit point condition for the linear relation T_{max }. In the limit circle case the Evans–Everitt condition is proved to hold on a subspace T_N^* of T_{max } characterized by the Neumann boundary condition at b. The notion of the principal Titchmarsh–Weyl coefficient of this integral system is introduced. Boundary triple for the linear relation T_{max } in the limit point case (and for T_{N}^* in the limit circle case) is constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh–Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of R and W and the formula relating the principal Titchmarsh–Weyl coefficients of the direct and the dual integral systems is proved. For every integral system with the principal Titchmarsh–Weyl coefficients q a canonical system is constructed so that its Titchmarsh–Weyl coefficient Q is the unwrapping transform of q: Q(z)=z q(z^2).

Second-Order Differential Operators in the Limit Circle Case

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

This paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.

Density of the spectrum of Jacobi matrices with power asymptotics

We consider Jacobi matrices $J$ whose parameters have the power asymptotics $\rho_n=n^{\beta_1} \left( x_0 + \frac{x_1}{n} + {\rm O}(n^{-1-\epsilon})\right)$ and $q_n=n^{\beta_2} \left( y_0 + \frac{y_1}{n} + {\rm O}(n^{-1-\epsilon})\right)$ for the off-diagonal and diagonal, respectively. We show that for $\beta_1 > \beta_2$, or $\beta_1=\beta_2$ and $2x_0 > |y_0|$, the matrix $J$ is in the limit circle case and the convergence exponent of its spectrum is $1/\beta_1$. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix $J$ have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case $\lim_{n\to \infty} |q_n|\big/ \rho_n = 2$) and determine the convergence exponent in almost all cases.

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

NONLINEAR SINGULAR STURM-LIOUVILLE PROBLEMS WITH IMPULSIVE CONDITIONS

In this paper, we consider a non-linear impulsive Sturm-Liouville problem on semiinfinite intervals in which the limit-circle case holds at infinity for THE Sturm-Liouville expression. We prove the existence and uniqueness theorems for this problem.

Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

In the Hilbert space $$L_{W}^{2}([a,b);E)$$ ( $$-\infty<a<b\le +\infty ,$$ $$\dim E=N<+\infty ,$$ $$W>0$$ ) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.

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.

Titchmarsh–Weyl theory for q-Dirac systems

In this work, we establish Titchmarsh–Weyl theory for singular [Formula: see text]-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to [Formula: see text]-analogue of this system. We show that it does not occur for the limit-circle case for the [Formula: see text]-Dirac system.

A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

In this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case.

Titchmarsh–Weyl Theory for Dirac Systems with Transmission Conditions

The present paper deals with singular Dirac systems with transmission conditions. We will define limit-point and limit-circle singularities. We will also show that it does not occur the limit-circle case for the Dirac systems with transmission conditions.

Spectral Analysis for Dissipative Singular Sturm–Liouville Operators with Transmission Conditions

In the present paper, we study a class of discontinuous dissipative differential operators in Weyl’s limit circle case. Using the characteristic determinant, we prove the completeness of the system of eigenvectors and associated vectors of this dissipative operator.