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

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.

7 pages, 5 figures, 1 table (v2: misprints fixed)

References in corpus (7)

• Perfect state transfer in quantum spin networks
• Spin Chains as Perfect Quantum State Mirrors
• Energy Transport in Trapped Ion Chains
• Exploiting boundary states of imperfect spin chains for high-fidelity state transfer
• Diagonalization of quasi-uniform tridiagonal matrices
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Cited by in corpus (5)

• Exponential quantum speedup in simulating coupled classical oscillators
• A classical model for perfect transfer and fractional revival based on $q$-Racah polynomials
• Dispersionless pulse transport in mass-spring chains: All possible perfect Newton's cradles
• Analytic "Newton's cradles" with perfect transfer and fractional revival
• A superintegrable quantum field theory