Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems

Abstract

We consider the energy dependent Schrödinger operator$\mathbb{L} = \sum\limits_{i = 0}^N {\lambda ^i (\varepsilon _i \partial ^2 + u_i )}$, which we have previously shown to be associated with multi-Hamiltonian structures [2]. In this paper we use an unusual form of the Lax approach to derive by asingle construction the time evolutions of the eigenfunctions of$\mathbb{L}$, the associated Hamiltonian operators and the Hamiltonian functionals. We then generalise the well known factorisation of standard Lax operators to the case of energy-dependent operators. The simple product of linear factors is replaced by a λ-dependent quadratic form. We thus generalise the resulting construction of Miura maps and modified equations. We show that for some of our systems there exists a sequence ofN such modifications, ther ^th modification possessing (N−r+1) Hamiltonian structures.