Abstract
In this paper we develop the spectral theory for discrete symplectic systems with general jointly varying endpoints. This theory includes a characterization of the eigenvalues, construction of the M-lambda function and Weyl disks, their matrix radii and centers, statements about the number of square summable solutions, and limit point or limit circle analysis. These results are new even in some particular cases, such as for the periodic and antiperiodic endpoints, or for discrete symplectic systems with special linear dependence on the spectral parameter. The method utilizes a new transformation to separated endpoints, which is simpler and more transparent than the one in the known literature.MSC: 39A12, 34B20, 34B05, 47B39.
Highlights
1 Motivation In this paper we develop the spectral theory, in particular the Weyl-Titchmarsh theory, for discrete symplectic systems zk+ (λ) = (Sk + λVk)zk(λ), (Sλ) in which the dependence on the spectral parameter λ ∈ C is linear but other than that general
Due to the augmented structure of the matrix R+(λ), which has dimension n, it is the rank of R+(λ) alone which determines the number of linearly independent square summable solutions of (Sλ)
5 Conclusion In this paper we demonstrated that n-dimensional discrete symplectic eigenvalue problems with jointly varying endpoints can be studied by a transformation to separated endpoints in the dimension n
Summary
The following result is quite surprising in the sense that one would expect to have R+(λ) = in the limit point case; see [ , Theorem . ]. To the contrary, due to the augmented structure of the matrix R+(λ), which has dimension n, it is the rank of R+(λ) alone which determines the number of linearly independent square summable solutions of (Sλ). We calculate the rank of the limiting matrix radius R+(λ) and compare it with the corresponding number of linearly independent square summable solutions.
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