Unitarity of the SoV Transform for the Toda Chain
Unitarity of the SoV Transform for the Toda Chain
复制标题
Toda 链 SoV 变换的统一性
DOI：
10.1007/s00220-014-2134-6
复制
发表时间：
2013
影响因子：
2.4
通讯作者：
K. Kozlowski
中科院分区：
文献类型：
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作者：
K. Kozlowski
量子分离变量方法包括将原始希尔伯特空间映射到一个频谱问题的频谱问题采取更简单的“分离”形式。为了实现这样一个程序，人们应该明确地构造映射，然后证明它是酉的。在本文中，我们开发了一种技术，它允许一个证明的情况下，量子户田链的这种映射的么正性。我们的证明仅仅建立在所谓的量子逆散射方法的框架中自然产生的对象和关系上。因此，通过微小的修改，它应该很容易转置到其他量子可积模型可解的量子分离变量方法。因此，它为基于量子户田链的群论解释的映射的么正性证明提供了一个重要的替代方案，而这对于更复杂的量子可积模型是不存在的。
The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler “separated” form. In order to realise such a program, one should construct the map explicitly and then show that it is unitary. In the present paper, we develop a technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the quantum separation of variables method. As such, it provides an important alternative to the proof of the map’s unitarity based on the group theoretical interpretation of the quantum Toda chain, which is absent for more complex quantum integrable models.