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首页> 外文期刊>Journal of geometry and symmetry in physics >ON SPECTRAL THEORY OF LAX OPERATORSON SYMMETRIC SPACES: VANISHING VERSUSCONSTANT BOUNDARY CONDITIONS
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ON SPECTRAL THEORY OF LAX OPERATORSON SYMMETRIC SPACES: VANISHING VERSUSCONSTANT BOUNDARY CONDITIONS

机译：松弛算子对称空间的谱理论：消失的对等常边界条件

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摘要

We outline several specific issues concerning the theory of multi-component nonlinear Schrodinger equations with vanishing and constant bound-ary conditions. We start with the spectral properties of the Lax operator L forvanishing boundary conditions. We introduce the fundamental analytic solutions(FAS) and demonstrate their importance for relating the scattering problem to aRiemann-Hilbert problem, and for the construction of the resolvent of L. Thenwe generalize this procedure to constant boundary conditions case. We start withthe structure of the class of allowed potentials M and give a recipe of how FAScan be constructed on each of the leafs of the relevant Riemannian surface. Thisallows us to relate the scattering problem to a Riemann-Hilbert problem posed ona Riemannian surface. Next we use these FAS to construct the resolvent of L andstudy its spectral properties. We also introduce the minimal set of scattering dataon the continuous spectrum of L which generically has varying multiplicity. Thegeneral construction is illustrated by three representative examples related to A.III,C.II and D.III symmetric spaces. Finally we consider regularized Wronskian re-lations which allow us to analyze the mapping between the potential of L and thescattering data.
机译：我们概述了有关具有消失和恒定边界条件的多分量非线性Schrodinger方程理论的几个具体问题。我们从消除边界条件的Lax算子L的光谱特性开始。我们介绍了基本解析解（FAS），并证明了其对于将散射问题与Aiemann-Hilbert问题相关联以及构造L的分解体的重要性。然后，将该过程推广到恒定边界条件的情况。我们从允许电位M类的结构开始，并给出如何在相关黎曼面的每个叶子上构造FAScan的方法。这使我们能够将散射问题与摆在黎曼曲面上的黎曼-希尔伯特问题联系起来。接下来，我们使用这些FAS构造L的分解物并研究其光谱特性。我们还介绍了L的连续光谱上的最小散射数据集，L的连续光谱通常具有变化的多重性。通过与A.III，C.II和D.III对称空间有关的三个代表性示例来说明一般构造。最后，我们考虑正则化的Wronskian关系，这使我们能够分析L的电势和散射数据之间的映射。

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