Sturmian and spectral theory for discrete symplectic systems

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Authors	BOHNER Martin DOŠLÝ Ondřej KRATZ Werner
Year of publication	2009
Type	Article in Periodical
Magazine / Source	Trans. Amer. Math. Soc.
MU Faculty or unit	Faculty of Science
Citation
Web	http://www.ams.org/tran/2009-361-06
Field	General mathematics
Keywords	Discrete symplectic system; discrete quadratic functional; Rayleigh principle; extended Picone identity
Description	We consider symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n. In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.
Related projects:	Mathematical structures and their physical applications Difference equations and dynamic equations on time scales II