Complex oscillations of non-definite sturm-liouville problems
Number of Authors: 2
2016 (English)In: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, 314Article in journal (Refereed) Published
We expand upon the basic oscillation theory for general boundary problems of the form (Formula presented) where q and r are real-valued piecewise continuous functions and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. The non-definite case is characterized by the indefiniteness of each of the quadratic forms (Formula presented) over a suitable space where B is a boundary term. In 1918 Richardson proved that, in the case of the Dirichlet problem, if r(t) changes its sign exactly once and the boundary problem is non-definite then the zeros of the real and imaginary parts of any non-real eigen-function interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigen-functions. Furthermore, we show that when a non-real eigen-function vanishes inside I, the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.
Place, publisher, year, edition, pages
Research subject Mathematics
IdentifiersURN: urn:nbn:se:ltu:diva-61277ISI: 000390155500003OAI: oai:DiVA.org:ltu-61277DiVA: diva2:1060621
Validerad; 2017; Nivå 2; 2017-01-12 (andbra)2016-12-292016-12-292017-01-12Bibliographically approved