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

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Authors

ŠIMON HILSCHER Roman ZEMÁNEK Petr

Year of publication 2014
Type Article in Periodical
Magazine / Source Applied Mathematics and Computation
MU Faculty or unit

Faculty of Science

Citation
Doi http://dx.doi.org/10.1016/j.amc.2013.12.135
Field General mathematics
Keywords Weyl-Titchmarsh theory; time scale; symplectic system; linear Hamiltonian system; limit point case; limit circle case; square integrable solution; coupled boundary conditions
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Description 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.
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