Spectral analysis of fourth order differential operators III

Abstract

This paper extends the results of the two previous papers in several directions. For one we allow slower decay of the coefficients, but higher order differentiability. For this an expansion for the diagonalizing transformations is derived. Secondly unbounded coefficients are permitted. This requires further transformations in order to achieve Levinson's form, but also a modification with the usual M-matrix approach. While the standard results carry over to weakly singular coefficients, very singular coefficients will generally lead to discrete spectra.

Spectral analysis of higher order differential operators with unbounded coefficients

“…Ways to extend these results to coefficients with even less regularity are presented in [1], [2], [3], [6], [9]. So we will not bother here with excessive generalities.…”

“…Asymptotic integration in the analysis of the absolutely continuous spectrum requires results which are uniform in the spectral parameter z. This has been developed by Remling, Hinton and one of the authors in a series of papers [1], [2], [3], [6], [9], [17]. In [17] fourth order operators with unbounded coefficients were analyzed.…”

“…Thus, one may add various perturbing terms to the coefficients. In earlier publications [1]- [3], [6], [9], we had coined the expression "Levinson terms" to mean all expressions which ultimately are transformed into L 1 -terms and thus being irrelevant for the asymptotics. Here, a Levinson term should signify an expression which is ultimately transformed into L 1 · x − .…”

“…From (3.17), the deficiency index can be read off more often or less directly, but (3.17) and (3.18) can also be used to determine the M -matrix and thus the location of the absolutely continuous spectrum [1], [3], [9], [17]. Then [ [16], Theorem 6.1] applied to T will also show the absence of singular continuous spectrum if (3.18) holds uniformly in K .…”

“…While the general method to diagonalize Λ + S has been described in [3], we will use a simplified transformation…”