Miura Type Transform Between Non-Abelian Volterra and Toda Lattices and Inverse Spectral Problem for Band Operators

Abstract

We study a discrete Miura-type transformation between the hierarcies of non-Abelian semi-infinite Volterra (Kac–van Moerbeke) and Toda lattices with operator coefficients in terms of the inverse spectral problem for three-diagonal band 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 operator-valued function, can be used in solving initial-boundary value problem for the systems of both these hierarchies. 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 hierarchy and the Toda sub-hierarchy which can be characterized via Lax operators corresponding to its lattices.