Let $a_k(t), b_k (t) \in \mathbb{R} \ (k = 1, 2, \dots, n)$ satisfy the following differential equations: $$\begin{aligned} \frac{d}{dt}a_k(t) &= 2 \left( b_k^2 - b_{k-1}^2 \right) \\ \frac{d}{dt} b_k(t) &= b_k \left( a_{k+1} - a_{k} \right)\end{aligned} \qquad k = 1, 2, \dots , n $$ where $b_0(t) = b_n(t) = 0$. Consider the $n \times n$ tri-diagonal matrix $$L(a,b)=\begin{bmatrix} a_1 & b_1 & 0 & \dots & 0 &0\\ b_1 & a_2 & b_2 & \dots & 0 &0\\ \vdots & \vdots & \vdots & \ddots & \vdots& \vdots \\ 0 & 0 & 0 & \dots & a_{n-1} & b_{n-1}\\ 0 & 0 & 0 & \dots & b_{n-1} & a_n \end{bmatrix}$$ show that:
- The eigenvalues of $L(t) = L(a(t), b(t))$ are independent of $t$.
- $\lim_{t \to \infty}b_k(t) = 0$ for $k=1,2,\dots,n-1$.
For P1, Let $L(a,b)=\begin{bmatrix} 0 & b_1 & 0 & \dots & 0 &0\\ -b_1 & 0 & b_2 & \dots & 0 &0\\ \vdots & \vdots & \vdots & \ddots & \vdots& \vdots \\ 0 & 0 & 0 & \dots & 0 & b_{n-1}\\ 0 & 0 & 0 & \dots & -b_{n-1} & 0 \end{bmatrix}$, then all above becomes $\frac{dL}{dt}=PL-LP$. Since $L$ exists and is unique, $P$ also exists and is unique. Hence there exists only one $U$ such that $\frac{d}{dt}U(t,s)=P(t)U(t,s),U(s,s)=I$. Since $U(t,s)L(s)U(t,s)^{-1}$ is also the solution to $\frac{dL}{dt}=PL-LP$, we must have $L(t)=U(t,s)L(s)U(t,s)^{-1}$ and P1 is solved.
However now I'm stucking on P2, which is equivalent to prove $\lim_{t\rightarrow \infty}P=0$ or $L$ tends to be diagonal, in which the solution of ODE tends to be constant, and I don't know how to handle with analyzing matrix ODE.
Any help, hint or solution on Problem 2 would be appreciated!
These are the equations of motion of the Toda lattice . It is an integrable classical mechanical system, with a Lax matrix given by $L(a,b)$ as you define. The eigenvalues of a Lax matrix can be shown to be time-independent, see here.