Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems
Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems
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DOI：
10.1007/bf01219659
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发表时间：
1989-09
影响因子：
2.4
通讯作者：
M. Antonowicz;A. Fordy
中科院分区：
文献类型：
--
作者：
M. Antonowicz;A. Fordy
我们考虑能量相关的薛定谔算子，我们之前已经证明它与多哈密顿结构相关[2]。在本文中，我们使用 Lax 方法的一种不寻常形式，通过单一构造导出相关哈密顿算子和哈密顿泛函的本征函数的时间演化。然后，我们将众所周知的标准 Lax 算子因式分解推广到依赖于能量的算子的情况。线性因子的简单乘积被 λ 相关的二次形式所取代。因此，我们概括了 Miura 映射和修改方程的结果构造。我们证明，对于我们的一些系统，存在一系列 N 个这样的修改，其中第一个修改具有 (N−r+1) 个哈密顿结构。
We consider the energy dependent Schrödinger operator, 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 constructionthe time evolutions of the eigenfunctions of, 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 ofNsuch modifications, therthmodification possessing (N−r+1) Hamiltonian structures.