The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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.

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Sturmian and spectral theory for discrete symplectic systems

We introduce the comparative index of two conjoined bases of a symplectic difference system, which generalizes difference analogs of canonical systems of differential equations. We consider the main properties of the comparative index and its relation to the number of focal points of a conjoined basis of the symplectic system. We prove a formula relating the number of focal points (including multiplicities) of two bases in the interval (i, i + 1] and corollaries of this formula such as an estimate for the difference of the numbers of focal points of two conjoined bases in the interval (0, N + 1], the equality of the numbers of focal points of principal solutions for the primal and reciprocal systems, sufficient conditions for the solvability of the Riccati equation for a disconjugate symplectic system, etc.

Comparative index for solutions of symplectic difference systems

Comparative index and Sturmian theory for linear Hamiltonian systems

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 $$
\hat W_i 
$$
 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.

Comparison theorems for symplectic systems of difference equations

The comparative index for conjoined bases of symplectic difference systems

Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index

This paper studies symplectic transformations for conjoined bases of the symplectic difference systems where the matrix W i is symplectic for We derive new relations between the number of focal points of Y i and P i Y i , where P i is an arbitrary symplectic transformation. We formulate conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems. In particular, for the case and , we establish duality between eventual disconjugacy of general symplectic systems and their reciprocals.

Transformations and the number of focal points for conjoined bases of symplectic difference systems

On relative oscillation theory for symplectic eigenvalue problems

This monograph is devoted to covering the main results in the qualitative theory of symplectic difference systems, including linear Hamiltonian difference systems and Sturm-Liouville difference equations, with the emphasis on the oscillation and spectral theory. As a pioneer monograph in this field it contains nowadays standard theory of symplectic systems, as well as the most current results in this field, which are based on the recently developed central object - the comparative index. The book contains numerous results and citations, which were till now scattered only in journal papers. The book also provides new applications of the theory of matrices in this field, in particular of the Moore-Penrose pseudoinverse matrices, orthogonal projectors, and symplectic matrix factorizations. Thus it brings this topic to the attention of researchers and students in pure as well as applied mathematics.

Julia Elyseeva

Papers

The comparative index for conjoined bases of symplectic difference systems

Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index

Transformations and the number of focal points for conjoined bases of symplectic difference systems

On relative oscillation theory for symplectic eigenvalue problems

Symplectic Difference Systems: Oscillation and Spectral Theory