Qualitative and Spectral theory of some regular non-definite Sturm-Liouville problems

Abstract: In this Licentiate thesis, we study some regular non-definite Sturm-Liouville problems. In this case, the weight function takes on both positive and negative signs on a given interval [a, b]. One feature of the non-definite Sturm-Liouville problem is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist.This thesis consists of three papers (papers A-C) and an introduction to this area, which puts these papers into a more general frame.In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper.In paper B we show that the interlacing property which holds in the one turning point case does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (−1, 2). We also present some theoretical results which support the numerical results.In paper C we extend results found in the paper by Jussi Behrndt et.al, in an essential way, to a case in which the weight function vanishes identically in a subinterval of [a, b]. In particular, we present some surprising numerical results on a specific problem in which the weight function is allowed to vanish identically on a subinterval of [−1, 2]. We also give some theoretical results which support these numerical examples.