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An Ablowitz-Ladik system with a discrete potential. I: Extended resolvent. (English. Russian original) Zbl 0960.37035
Theor. Math. Phys. 119, No. 1, 407-419 (1999); translation from Teor. Mat. Fiz. 119, No. 1, 20-33 (1999).
This paper is devoted to the spectral theory of the operator \(L(w)\) given by the infinite matrix \[ L_{m,n} (w)= \delta_{m,n-1}- \begin{pmatrix} w &r_n\\ s_n &1/w \end{pmatrix} \delta_{m,n}, \quad m,n\in \mathbb{Z},\;w\in \mathbb{C}, \] where each element of the matrix is a \((2\times 2)\) matrix, \(\delta_{m,n}\) is the Kronecker symbol, and a \((2\times 2)\) unit matrix factor is omitted in the term \(\delta_{m,n-1}\) and \(r_n,s_n\in \{0,1\}\), \(n\in \mathbb{Z}\). The authors consider potentials with finite support and corresponding an Ablowitz-Ladik \(L(w) \varphi=0\) linear system, which is known to be the discretized version of the Zakharov-Shabat linear problem. Here the extended resolvent of this system is constructed and the singularities of this operator are analyzed in detail.
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
47B39 Linear difference operators
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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