Petr Zemánek

On a Weyl–Titchmarsh theory for discrete symplectic systems on a half line

Recently, Bohner and Sun introduced basic elements of a Weyl-Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl-Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Greens function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy lambda-dependent boundary conditions at infinity.

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.

Limit point and limit circle classification for symplectic systems on time scales

Abstract In this paper we study the limit point and limit circle classification for symplectic systems on time scales, which depend linearly on the spectral parameter. In a broader context, we develop a unified Weyl–Titchmarsh theory for continuous and discrete linear Hamiltonian and symplectic systems. Both separated and coupled boundary conditions are allowed. Our results include the study of the Weyl disks and circles and their limiting behavior, as well as a precise analysis of the number of linearly independent square integrable solutions. We also prove an analogue of the famous Weyl alternative. We connect and unify many known results in the Weyl–Titchmarsh theory for continuous, discrete, and special time scales systems and explain the differences between them. Some of our statements, in particular those connected with coupled endpoints or the Weyl alternative, are new even in the continuous time setting.

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. Zemanek (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.

Weyl disks and square summable solutions for discrete symplectic systems with jointly varying endpoints

In this paper we develop the spectral theory for discrete symplectic systems with general jointly varying endpoints. This theory includes a characterization of the eigenvalues, construction of the M-lambda function and Weyl disks, their matrix radii and centers, statements about the number of square summable solutions, and limit point or limit circle analysis. These results are new even in some particular cases, such as for the periodic and antiperiodic endpoints, or for discrete symplectic systems with special linear dependence on the spectral parameter. The method utilizes a new transformation to separated endpoints, which is simpler and more transparent than the one in the known literature.MSC: 39A12, 34B20, 34B05, 47B39.

A note on the equivalence between even order Sturm–Liouville equations and symplectic systems on time scales☆

Abstract The 2 n th -order Sturm–Liouville differential and difference equations can be written as linear Hamiltonian differential systems and symplectic difference systems, respectively. In this work, a similar result is given for the 2 n th -order Sturm–Liouville equation on time scales using time reversed symplectic dynamic systems. Moreover, we show that this transformation preserves the value of the corresponding quadratic functionals which enables a further generalization of the theory for continuous and discrete Sturm–Liouville equations.

On discrete symplectic systems: Associated maximal and minimal linear relations and nonhomogeneous problems

In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental properties of the corresponding deficiency indices, including a relationship between the number of square summable solutions and the dimension of the defect subspace, are also derived. Moreover, a sufficient condition for the existence of a densely defined operator associated with the symplectic system is provided.

Time scale symplectic systems with analytic dependence on spectral parameter

This paper is devoted to the study of time scale symplectic systems with polynomial and analytic dependence on the complex spectral parameter . We derive fundamental properties of these systems (including the Lagrange identity) and discuss their connection with systems known in the literature, in particular with linear Hamiltonian systems. In analogy with the linear dependence on , we present a construction of the Weyl disks and determine the number of linearly independent square integrable solutions. These results extend the discrete time theory considered recently by the authors. To our knowledge, in the continuous time case this concept is new. We also establish the invariance of the limit circle case for a special quadratic dependence on and its extension to two (generally nonsymplectic) time scale systems, which yields new results also in the discrete case. The theory is illustrated by several examples.

Generalized Lagrange Identity for Discrete Symplectic Systems and Applications in Weyl–Titchmarsh Theory

In this paper we consider discrete symplectic systems with analytic dependence on the spectral parameter. We derive the Lagrange identity, which plays a fundamental role in the spectral theory of discrete symplectic and Hamiltonian systems. We compare it to several special cases well known in the literature. We also examine the applications of this identity in the theory of Weyl disks and square summable solutions for such systems. As an example we show that a symplectic system with the exponential coefficient matrix is in the limit point case.

Characterization of self-adjoint extensions for discrete symplectic systems

All self-adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized. Especially, for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given expression are discussed and the Krein--von Neumann extension is described explicitly. In addition, a limit point criterion for symplectic systems is established. The result partially generalizes even the classical limit point criterion for the second order Sturm--Liouville difference equations.