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分类： Differential equations

  • Sturm-Liouville theory 与周期轨道：谱分解、边界条件和 monodromy

    旧博客原文

    原题：Periodic orbits and Sturm–Liouville theory

    I thinks there is some problem related to the solution of a 2 order differential equation given by Sturm-Liouville system which is nontrivial.

    It is well-know that the power of Sturm-Liouville theory see  wiki, is due to it is some kind of “spectral decomposition” in the solution space.

    Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURM-LIOUVILLE. PROBLEMS BY APPROXIMATING THE DIFFERENTIAL EQUATION.

    I post two problem here, this is a product due to a random walk along the boundary of topology and the analysis,

    Problem 1.

    Fix a set A=\{k_1<k_2<...<k_l\}, is there a 2 order ordinary differential equation given by Sturm–Liouville theory  such that the eigenfunction f_{k} is periodic if and only if k\in A?

    There is also some weak version of this and a infinity version of this.

    Of course we have the following map, from the high order ordinary differential equation to the 1 order differential equation in high dimension. But the key point is that it is not a bijection! The Frobenius condition play a crucial role.

    Problem 2.

    There is a homotopy in the moduli space of differential equation, and we could define a direct product operator in this space, and we consider the topology defamation of the eigenfunction, could there be some equality, one side of it explain the topology information, the other side explain the spectral (or analysis) information?

    There is another interesting problem.

    补充说明

    以下是新整理的中文说明；上方旧博客原文保持不变。

    Sturm-Liouville theory 把二阶线性微分方程变成谱问题。若再加入周期边界条件，就会自然出现 monodromy matrix、周期解和一维动力系统的交界。

    Sturm-Liouville theory 与周期轨道：谱分解、边界条件和 monodromy
    Sturm-Liouville 周期谱可以通过一阶系统的 monodromy matrix 来刻画。

    1. Sturm-Liouville 系统

    标准形式为

    $$-(p(x)y’)’+q(x)y=\lambda w(x)y.$$

    在合适边界条件下，这是自伴特征值问题，特征函数构成正交基。

    2. 周期边界条件

    若区间为 $[0,T]$，周期解满足

    $$y(0)=y(T),\qquad y'(0)=y'(T).$$

    这等价于一阶系统的 monodromy matrix 有特征值 $1$。

    3. 从二阶到一阶系统

    令 $Y=(y,py’)$，二阶方程可写为

    $$Y’=A_\lambda(x)Y.$$

    基本解矩阵 $M_\lambda(T)$ 描述一个周期后的变化。周期谱由

    $$\det(M_\lambda(T)-I)=0$$

    刻画。

    4. 反问题

    一个自然问题是：能否指定某个集合 $S$，使得周期特征值恰好落在 $S$ 中？这类问题接近 inverse spectral theory，需要理解势函数 $q(x)$ 如何控制 monodromy。

    5. 拓扑与分析

    周期轨道是动力系统对象，Sturm-Liouville 是谱分析对象。monodromy matrix 把二者联系起来：谱参数变化时，monodromy 在矩阵群中运动，周期解对应它穿过特定子集。