Asymptotic formulas for discrete eigenvalue problems in Liouville normal form

In the analysis of both continuous and discrete eigenvalue problems, asympt otic formulas play a central and crucial role. For example, they have been fundamental in the derivation of results about the inversion of the free os cillation problem of the Earth and related inverse eigenvalue problems, the computation of uniformly valid eigenvalues approximations, the proof of re sults about the behavior of the eigenvalues of Sturm-Liouville problems wit h discontinuous coefficients, and the construction of a counterexample to t he Backus-Gilbert conjecture. Useful formulas are available for continuous eigenvalue problems with general boundary conditions as well as for discret e eigenvalue problems with Dirichlet boundary condition. The purpose of thi s paper is the construction of asymptotic formulas for discrete eigenvalue problems with general boundary conditions. The motivation is the computatio n of uniformly valid eigenvalue approximations. It is now widely accepted t hat the algebraic correction procedure, first proposed by Paine ct aL,(13) is one of the simplest methods for computing uniformly valid approximations to a sequence of eigenvalues of a continuous eigenvalue problem in Liouvil le normal form.(8) This relates to the fact that, for Liouville normal form s with Dirichlet boundary conditions, it is not too difficult to prove that such procedures yield, under quite weak regularity conditions, uniformly v alid O(h(2)) approximations. For Liouville normal forms with general bounda ry conditions, the corresponding error analysis is technically more challen ging. Now it is necessary to have, for such Liouville normal forms, higher order accurate asymptotic formulas for the eigenvalues and eigenfunctions o f their continuous and discrete counterparts. Assuming that such asymptotic formulas are available, it has been shown(1) haw uniformly valid O(h(2)) r esults could be established for the application of the algebraic correction procedure to Liouville normal forms with general boundary conditions. Algo rithmically, this methodology represents an efficient procedure for determi ning uniformly valid approximations to sequences of eigenvalues, even thoug h it is more complex than for Liouville normal forms with Dirichlet boundar y conditions. As well as giving a brief review of the subject for general ( Robin) boundary conditions, this paper sketches proofs for the asymptotic f ormulas, for Robin boundary conditions, which are required in order to cons truct the mentioned O(h2) results.