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Discrete symplectic systems, boundary triplets, and self-adjoint extensions

Volume 579 / 2022

Petr Zemánek, Stephen L. Clark Dissertationes Mathematicae 579 (2022), 1-87 MSC: Primary 47A06; Secondary 47A20, 39A70, 47B39, 39A12. DOI: 10.4064/dm838-12-2021 Published online: 5 May 2022


An explicit characterization of all self-adjoint extensions of the minimal linear relation associated with a discrete symplectic system is provided using the theory of boundary triplets with special attention paid to the quasiregular and limit point cases. A particular example of the system (the second order Sturm–Liouville difference equation) is also investigated thoroughly, while higher order equations or linear Hamiltonian difference systems are discussed briefly. Moreover, the corresponding gamma field and Weyl relations are established and their connection with the Weyl solution and the classical $M(\lambda)$-function is discussed. To make the paper reasonably self-contained, an extensive introduction to the theory of linear relations, self-adjoint extensions, and boundary triplets is included.


  • Petr ZemánekDepartment of Mathematics and Statistics
    Faculty of Science
    Masaryk University
    Kotlářská 2
    CZ-61137 Brno, Czech Republic
  • Stephen L. ClarkDepartment of Mathematics & Statistics
    101 Rolla Building
    Missouri University of Science and Technology
    Rolla, MO 65409-0020, USA

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