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

TL;DR

This paper introduces a specific persymmetric Jacobi matrix with square-integer eigenvalues, useful for modeling dispersionless, perfectly periodic mass-spring chains and testing numerical algorithms.

Contribution

It provides explicit formulas for a Jacobi matrix with square-integer eigenvalues and links it to the dynamics of dispersionless mass-spring chains.

Findings

Eigenvalues are 2k^2 for k=0,...,n-1.

Matrix entries are explicit functions of n.

Chain dynamics can be perfectly periodic and dispersionless.

Abstract

A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$ $(k=0, ..., n-1)$ 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.