Exact solutions for Sturm–Liouville problems on trees via novel substitute systems and the Wittrick–Williams algorithm

Abstract

Since 1970, the Wittrick–Williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately. The underlying theorem enables its application to any discipline requiring eigenvalues of self-adjoint systems of differential equations. Its value in mathematics was recently illustrated by studying Sturm–Liouville equations on large homogeneous trees with Dirichlet boundary conditions and n (≤43) levels. Recursive subsysteming was applied n −1 times to assemble the tree progressively from sub-trees. Hence, numerical results confirmed the recent theoretical bounds of Sobolev & Solomyak for n →∞. In addition, a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b . Inspired by the structural mechanics analogy, we now prove that all eigenvalues of the tree are obtainable from n substitute chains r (=1, 2, …,  n ) which involve only r consecutively linked differential equations and which have only singlefold eigenvalues. Equations are also derived for the multiplicities these eigenvalues have for the tree. Hence, double precision calculations on a PC readily gave eigenvalues for n =10 6 and b =10, i.e. ≃10 999 999 linked Sturm–Liouville equations. Moreover, a simple equation is derived which gives all the eigenvalues of uniform trees with Dirichlet conditions at both ends, and band-gap spectra are numerically demonstrated and theoretically justified for trees with the Dirichlet conditions at either end replaced by Neumann ones. Additionally, even if each multiple eigenvalue would be counted as if it were singlefold, the proportion of eigenvalues that are multiple is proved to approach unity as n →∞.