Let $r\geq 1$ be a positive integer and $M(x)$ be a $r\times r$ matrix defined by $M_{jl}(x)=(\partial_x)^j \cos(\omega^l x)$ where $0\leq i,j\leq r-1$, $\omega=e^{\pi i/r}$. For example, for $r=3$, $$M(x)=\begin{pmatrix}\cos x&\cos\omega x &\cos\omega^2x\\ -\sin x &-\omega\sin\omega x &-\omega^2\sin\omega^2 x \\ -\cos x&-\omega^2\cos\omega x &-\omega^4\cos\omega^2x \end{pmatrix}.$$ Let $k_r$ be the smallest positive root of the equation $\det M(x)= 0$. For example, $k_1=\pi/2$, $k_2$ is the smallest positive solution to $\tan x+\tanh x=0$ (between $\pi/2$ and $\pi$), $k_3=\pi$...
Question: Find $\lim_{r\to\infty}k_r/r$.
Might be a useful hint: It is easy to prove that if $x$ is a root, then $\omega x$ is also a root, since we can prove that $\det M(\omega x) =\omega^{r(r-1)/2} \det M(x) $.
Background: The problem arises from a quantum mechanics context (or, a functional minimization problem): I'm trying to find the ground state wavefunction of the Hamiltonian $H=\hat{p}^{2r}$ in an infinite potential well $-1\leq x\leq 1$, where $\hat{p}=-i\partial_x$. The wavefunction is $$\psi(x)=\sum_{j=0}^{r-1}A_j\cos\omega^j k x,$$ where $A_j$ and $k$ are constants determined by $\partial_x^j\psi(x)|_{x=1}=0, 0\leq j\leq r-1$, and a normalization condition. Then the ground state energy is $E_r=k_r^{2r}$. I basically want to find a good asymptotic form of $E_r$ for large $r$.