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On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems. (English) Zbl 1065.39027

The paper can be viewed as a natural continuation of the authors’ recent works (2001-2002) on matrix-valued Schrödinger and Dirac type operators to discrete Hamiltonian systems (i.e Hamiltonian systems of difference equations). These investigations are part of a large program which includes
(i) a systematic asymptotic expansion of Weyl-Titchmarsch matrices and Green matrices as the spectral parameter tends to infinity;
(ii) the derivation of trace formulae for such systems;
(iii) the proof of certain uniqueness theorms (including Borg and Hochstadt-type theorems) for operations;
(iv) the applications of these results to related integrable systems.
The authors develop the basic theory of matrix-valued Weyl-Titchmarsch \(M\)-functions and the associated Green matrices for whole-line and half-line self-adjoint Hamiltonian finite difference systems with separated boundary conditions.

MSC:

39A12 Discrete version of topics in analysis
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
34B20 Weyl theory and its generalizations for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
39A70 Difference operators
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