# An inverse eigenvalue problem on Jacobi matrices

I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix with basis elements $|1\rangle$ to $|N\rangle$, it should be capable of generating the evolution $|\langle N|e^{-iHt}|1\rangle|^2=1$ for some time $t$. However, I'm also interested in trying to make the system have the property $|\langle 1|e^{-iHt'}|1\rangle|^2=0$ at regular intervals in the intervening times (and specifically, how small that interval can be).

I've reduced this problem to a question about the eigenvalues $\lambda_n$ of the matrix, for $n=1,\dots,N$, such that $\lambda_N > \lambda_{N-1} > \dots > \lambda_1 =0$. If $n$ is even $\lambda_n$ is an even integer and if $n$ is odd, $\lambda_n$ is an odd integer. We define

$R_k=\sum_{\lambda_n \equiv k \ \rm{ mod }\ M}\frac{(-1)^n}{B'(\lambda_n)}$

for $k=0,\dots,M-1$ for some integer $M$, and where $B'(\lambda)$ is the derivative of the function
$B(\lambda)=\prod_{n=1}^N(\lambda-\lambda_n)$ with respect to $\lambda$.

I need to determine what the maximum value of $M$ is such that all the $R_k$ can be equal, and would like an example of the set of eigenvalues that achieves this. If this is not possible, are there any examples of sets of eigenvalues such that the $R_k$ are all equal for $M>2$?

In this context, $\langle a|H|b\rangle$ denotes the matrix element $H_{ab}$.

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