Consider the following ODE on $\mathbb{C}^n$: \begin{equation} X''(t) = R(t) X(t), \end{equation} where $X(t)$ is a $n\times n$ complex-valued matrix and $R(t)$ has the following properties:positive definite, symmetric, smooth components.
I would like to obtain an estimate for $||X(t)||$ of the following kind
\begin{equation} e^{\sqrt{\lambda_{\min}}t} \lesssim \|X(t)\| \lesssim e^{\sqrt{\lambda_{\max}}t}, \end{equation} where the constants depend on the initial values $X(0), X'(0)$ and $\lambda_{\min},\lambda_{\max}$ are the positive square roots of the minimum and maximum eigenvalue of $R(t)$, uniformly in time. Precisely
\begin{equation} \lambda_{\min} = \min_{t\in [0,T]} \lambda_i(t), \quad \lambda_i(t) \in \sigma(R(t)),\: i=1,\ldots,n, \end{equation} $\sigma$is the spectrum. The same goes for $\lambda_{\max}$.
Note: I already proved the upper bound. I am not sure the lower bound holds with the positive sign. Estimates with $e^{-\lambda_{\min}t},e^{-\lambda_{\max}t}$ are OK as well.
Note(bis): I am not completely sure the lower bound holds in general but I was unable to provide a counterexample, so if you have one, that would be OK as well.
EDIT: as mentioned by @loup blanc, the result does not hold in general. Indeed the lower bound fails to be true. I encountered a result (Katchalov, Inverse spectral boundary problems Lemma 2.56) that gives conditions under which X(t) is non degenerate. Precisely, in addition to what I have assumed above, $X'(0)(X(0))^{-1}$ has to be symmetric and with positive definite imaginary part. The results fails to give a quantitative bound, which is exactly what I am after.
We assume $t>0$. I agree with the superior bound in the form $ \|X(t)\| \leq Ce^{\sqrt{\lambda_{\max}}t}$. Yet, the inferior bound may be $0$. See for example, the case $n=1,R=1$ and $X(t)=e^t-2e^{-t}$.
EDIT. Answer to @MBerra. You cannot obtain a lower bound that depends on a sole eigenvalue.
To see that, consider the case $n=2$ and $R$ is the constant matrix $diag(a^2,b^2)$ where $a\not=b$. Then $X(t)=(ue^{at}+ve^{-at},pe^{bt}+qe^{-bt})$ and $||X(t)||^2=|ue^{at}+ve^{-at}|^2+|pe^{bt}+qe^{-bt}|^2$; the minimum in $t\geq 0$, depends on $a,b$; moreover there is no formula for a lower bound that depends only on $a$ or $b$.