Asymptotic solutions and error estimates for linear systems of difference and differential equations

Abstract

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k + o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.

Asymptotic analysis of solutions of a radial Schrödinger equation with oscillating potential

“…We wish to emphasize that methods used in the proofs of Theorem 3.4 and to some extent also of Theorem 3.5 given below are similar to the methods used in [1], but these results do not immediately apply. The proof of Theorem 3.4 will be given in Appendix A.…”

“…For analyzing the solution of the scalar differential equation (1.1), it will be important that an induction argument shows that P r (t) are strictly lower triangular for all 1 ≤ r ≤ m − 1 because of the lower triangular structure of A 1 and the strictly lower triangular structure of B (1) r , and that P r (t) are lower triangular for all m ≤ r ≤ 2m − 1. For example, (3.14) with F j = 0 for all j shows that the P r 's are strictly lower triangular for 1 ≤ r ≤ m, and it follows that the P r (t)'s are strictly lower triangular for 1 ≤ r ≤ m − 1.…”

“…where K (1) 1 is a constant matrix and F are diagonal matrices whose entries that are complex conjugates of each other. To reduce…”

“…Then a fundamental result due to N. Levinson [8] could be used to obtain asymptotic representations for solutions. However, we will use a recent result [1] that requires a somewhat stronger hypothesis but in return provides more quantitative information on the error term.…”

On asymptotic summation of potentially oscillatory difference systems

“…In contrast the amount of works on asymptotic summation of difference systems is relatively small. See [2][3][4]8,11]. See also recent works [5] for a more general setting.…”