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On the completely indeterminate case for block Jacobi matrices

100%
Osipov A.
Concrete Operators
|
2017
|
tom 4
|
nr 1
48-57
EN
We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.
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Reconstruction of potential and boundary conditions for second order difference equations

88%
Currie S., Love A.
Commentationes Mathematicae
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2018
|
tom 58
|
nr 1-2
EN
Assume the eigenvalues and the weights are given for a difference boundary value problem and that the form of the boundary conditions at the endpoints is known. In particular, it is known whether the endpoints are fixed (i.e. Dirichlet or non-Dirichlet boundary conditions) or whether the endpoints are free to move (i.e. boundary conditions with affine dependence on the eigenparameter). This work illustrates how the potential as well as the exact boundary conditions can be uniquely reconstructed. The procedure is inductive on the number of unit intervals. This paper follows along the lines of S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, \emph{Quaestiones Mathematicae}, 40 (2017), no. 7, 861−877. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in the above reference, an additional spectrum is required more often than was the case in the unique reconstruction of the potential alone.
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