Inverse spectral problem for Jacobi operators and Miura transformation

Abstract

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.

Keywords

• jacobi operators
• positive operators
• deficiency indices
• power moment problem
• toda lattices
• volterra lattices
• miura transformations
• 44a60
• 47b36
• 37k10
• 37k15