Kratz, Werner; Šimon Hilscher, Roman; Zeidan, Vera Michel Eigenvalue and oscillation theorems for time scale symplectic systems. (English) Zbl 1218.34039 Int. J. Dyn. Syst. Differ. Equ. 3, No. 1-2, 84-131 (2011). This is a very comprehensive paper dealing with oscillation and spectral theory of symplectic dynamic systems on a time scale\[ x^{\Delta}=A(t)x+B(t)u,\quad u^{\Delta}=C(t)x+D(t)u, \tag{*} \]where the matrices \(A,B,C,D\) satisfy identities implying that the fundamental matrix of (*) is symplectic. The main result of the paper is the so-called oscillation theorem which relates the number of finite eigenvalues of a certain eigenvalue problem associated with (*) and the number of generalized focal points of a conjoined basis of this eigenvalue problem. To prove this result, the authors develop systematically the general theory of symplectic eigenvalue problems without any controllability assumption. The results of the paper cover as a particular case oscillation theorems for Hamiltonian differential systems (continuous case) and symplectic difference systems (discrete case). Reviewer: Ondřej Došlý (Brno) Cited in 12 Documents MSC: 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34N05 Dynamic equations on time scales or measure chains 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 34L05 General spectral theory of ordinary differential operators 34A30 Linear ordinary differential equations and systems Keywords:time scale symplectic system; linear Hamiltonian system; finite eigenvalue; generalized focal point; oscillation theorem PDFBibTeX XMLCite \textit{W. Kratz} et al., Int. J. Dyn. Syst. Differ. Equ. 3, No. 1--2, 84--131 (2011; Zbl 1218.34039) Full Text: DOI