Comparison theorems for symplectic systems of difference equations |
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Authors: | Yu V Eliseeva |
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Institution: | 1.Moscow State University,Moscow,Russia |
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Abstract: | In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference
analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of
conjoined bases of two symplectic systems with matrices W
i
and $
\hat W_i
$
\hat W_i
as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider
applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number
of eigenvalues λ of a symplectic boundary value problem on the interval (λ
1, λ
2]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined
bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes
the well-known reciprocity principle for discrete Hamiltonian systems. |
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Keywords: | |
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