A formula for the spectra of differential operators on graphs (Q366059)

This paper studies the eigenvalue problem of a second-order differential operator on a finite metric graph, with conditions on the first derivatives at the vertices ensuring self-adjointness. The main result is an explicit formula for the entire function \(\prod_{k=1}^\infty \Big(1-\frac{\lambda^2}{\lambda_k^2} \Big)\) which vanishes at the nonzero eigenvalues. The formula involves trigonometric polynomials in \(\lambda\) and is given in terms of graphs associated to a self-mapping of a finite set, introduced by V. I. Arnold.