Weyl Circles Research Articles

Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrodinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.

Weyl Function of Sturm–Liouville Problems with Transmission Conditions at Finite Interior Points

In this paper, a class of singular Sturm–Liouville problems with transmission conditions at finite interior points is studied. The equation, center and radius of Weyl circle are given. The necessary and sufficient conditions for the function $$ m(\lambda ) $$ , which is in the Weyl circle, are established. The definition of Weyl function for singular Sturm–Liouville problems with transmission conditions in the case of limit circle is given, and the properties of Weyl function are discussed.

Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on spectral parameter

In this paper we develop the Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. Our new theory includes characterizations of the Weyl discs and Weyl circles, their limiting behaviour, properties of square summable solutions including the analysis of the exact number of linearly independent square summable solutions and limit point/circle criteria. Some illustrative examples are also provided.

Weyl‐Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

We develop the Weyl‐Titchmarsh theory for time scale symplectic systems. We introduce the M(λ)‐function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl‐Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second‐order Sturm‐Liouville equations on time scales.

Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems on Sturmian time scales

We study non-self-adjoint Hamiltonian systems on Sturmian time scales, defining Weyl–Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterize realizations of the corresponding dynamic operator and its adjoint, and construct their resolvents. Even-order scalar equations and the Orr–Sommerfeld equation on time scales are given as examples illustrating the theory, which are new even for difference equations. These results unify previous discrete and continuous theories to dynamic equations on Sturmian time scales.

On M-functions and operator theory for non-self-adjoint discrete Hamiltonian systems

We study discrete, generally non-self-adjoint Hamiltonian systems, defining Weyl–Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterise realisations of the corresponding differential operator and its adjoint, and construct their resolvents.

Geometric Characterization of Singular Self-Adjoint Boundary Conditions for Hamiltonian Systems

The self-adjoint realizations with separated boundary conditions of rather general differential expressions on an interval [a, b) with a regular and b singular are investiga.ted. The permissible boundary conditions at the singular endpoint are first described in terms of the square integrable solutions of the associated differential equation. Then, for arbitrary (equal) deficiency indices, they are characterized using the M-functions lying on the boundary of the limiting object of the Weyl circles, thus generalizing a well-known result for Sturm-Liouville operators.

On the Operator-Valued Nevanlinna-Pick Problem

In this note we construct a specific Schur-Nevanlinna type algorithm that will be used in describing more precisely the solutions of the operator-valued Nevanlinna-Pick problem with the so-called method of Weyl circles. We also present an approach to the Pick criterion in the operator-valued setting.

A numerical method for the determination of the Titchmarsh-Weyl m -coefficient

A numerical method for determining the Titchmarsh-Weyl m ( λ ) function for the singular eigenvalue equation – ( py' )' + qy = λwy on [ a ,∞), where a is finite, is presented. The algorithm, based on Weyl’s theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case these circles ‘nest’ and tend to the limit-point m ( λ ). Using Weyl’s result for the diameter of the circles, error estimates for m ( λ ) are obtained. In 1971, W. N. Everitt obtained an extension of an integral inequality of Hardy-Littlewood, namely the help inequality. He showed that the existence of that inequality is determined by the properties of the null set of Im[ λ 2 m ( λ )]. In view of the major difficulties in analysing m ( λ ) even in the rare cases when it is given explicitly, very few examples of the help inequality are known. The computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities.

On the location of the Weyl circles

SynopsisThe paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.