From Toda to KdV
From Toda to KdV
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从户田到KdV
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
T. Paul
中科院分区：
文献类型：
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作者：
D. Bambusi;T. Kappeler;T. Paul
对于具有大量N个粒子的周期性户田链，我们考虑N−2-接近平衡的状态，并通过离散任意给定的网格尺寸为N−1的C2−函数来构造。我们的目标是描述出现在这些状态的动力学的Lax对公式中的Jacobi矩阵LN的谱，作为N → ∞。为此，我们构造了两个Hill算子H±--此类算子出现在Korteweg-de弗里斯方程的Lax对公式中--并用半经典分析方法证明，LN谱边缘特征值的N → ∞渐进性具有以下形式：其中是H±的特征值。在谱的大部分中，本征值是o（N−2）-接近平衡矩阵的本征值。作为一个应用程序，我们获得了类似类型的判别式，与LN的渐近。
For periodic Toda chains with a large number N of particles we consider states which are N−2-close to the equilibrium and constructed by discretizing arbitrary given C2−functions with mesh size N−1. Our aim is to describe the spectrum of the Jacobi matrices LN appearing in the Lax pair formulation of the dynamics of these states as N → ∞. To this end we construct two Hill operators H±—such operators come up in the Lax pair formulation of the Korteweg–de Vries equation—and prove by methods of semiclassical analysis that the asymptotics as N → ∞ of the eigenvalues at the edges of the spectrum of LN are of the form where are the eigenvalues of H±. In the bulk of the spectrum, the eigenvalues are o(N−2)-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to LN.