On computation of eigenvalues for differential equations by Newton’s method

Abstract:

Professor Zeev Nehari, in the last months before his untimely death, had been analyzing and developing a new algorithm for computing eigenvalues of selfadjoint boundary value problems of arbitrary order. Apparently, his main goals were Theorems 3.1 and 3.2 below (which incidentally yield the eigenvalues of the given problem, as well as those of a related problem). Unfortunately, as far as we know, Professor Nehari has not left a proof of the basic Lemma 3.1 which presumably was to be based on $\S 2$ or related results. It seems, however, worthwhile to publish this paper since the results of $\S \S 1$ and $2$ are complete and of independent interest and since the question of the validity of Lemma 3.1 is also of interest. While the presentation in $\S 1$ and $2$ is in most respects complete, there are a few places where some minor clarifications by selected footnotes were felt to be desirable. Also, Professor Nehari’s original incomplete $\S 3$ has been replaced by a revised and abbreviated version.