On unbounded commuting Jacobi operators and some related issues

We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.

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Deficiency indices of block Jacobi matrices and Miura transformation

Osipov

2022

Special Matrices

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We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.

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On unbounded commuting Jacobi operators and some related issues

Cited by 3 publications

References 13 publications

Inverse Spectral Problem for Band Operators and Their Sparsity Criterion In Terms of Inverse Problem Data

Inverse Spectral Problem for Band Operators and Their Sparsity Criterion In Terms of Inverse Problem Data

Inverse spectral problem for Jacobi operators and Miura transformation

Deficiency indices of block Jacobi matrices and Miura transformation

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