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

Abstract

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.

Keywords::

• Weyl–Titchmarsh theory
• symplectic system
• Weyl disc
• limit point case
• limit circle case
• square summable solution

Keywords::

• Primary 39A12
• Secondary 34B20
• 34B05
• 47B39

Acknowledgements

This research was supported by the Czech Science Foundation under grant P201/10/1032 and by the European Social Fund and the state budget of the Czech Republic under the project ‘Employment of Newly Graduated Doctors of Science for Scientific Excellence’ (registration number CZ.1.07/2.3.00/30.0009). The authors thank an anonymous referee whose comments and suggestions helped improve the presentation of the results. In particular, the authors appreciate his/her remarks, which led them to the statements in Lemma 2.2 and Theorem 4.17.

Notes

^1. Email: [email protected]