Persymmetric Jacobi matrices with square-integer eigenvalues and dispersionless mass-spring chains

A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$ $(k=0,\mathrm{.}\mathrm{.}\mathrm{.},n-1)$ is presented, with entries given as explicit functions of $n$. Besides the possible use for testing forward and inverse numerical algorithms, such a matrix is especially relevant for its connection with the dynamics of a mass-spring chain, which is a multi-purpose prototype model. Indeed, the mode frequencies being the square roots of the eigenvalues of the interaction matrix, one can shape the chain in such a way that its dynamics be perfectly periodic and dispersionless.