Sturmian and Spectral Theory for Discrete Symplectic Systems
We consider 2n x 2n 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.
M. Bohner et al., "Sturmian and Spectral Theory for Discrete Symplectic Systems," Transactions of the American Mathematical Society, American Mathematical Society, Jan 2009.
The definitive version is available at https://doi.org/10.1090/S0002-9947-08-04692-8
Mathematics and Statistics
Keywords and Phrases
discrete symplectic system; disrete quadratic functional; Sturmian separation result; Sturmian comparison result; Rayleigh principle; extended Picone identity
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