Electron. J. Differential Equations,
Vol. 2016 (2016), No. 314, pp. 17.
Complex oscillations of nondefinite SturmLiouville problems
Mervis Kikonko, Angelo B. Mingarelli
Abstract:
We expand upon the basic oscillation theory for general boundary problems
of the form
where q and r are realvalued piecewise continuous functions
and y is required to satisfy a pair of homogeneous separated boundary
conditions at the endpoints.
The nondefinite case is characterized by the indefiniteness of
each of the quadratic forms
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
nondefinite then the zeros of the real and imaginary parts of any nonreal
eigenfunction 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 nonreal eigenfunctions. Furthermore,
we show that when a nonreal eigenfunction vanishes inside I, the absolute
value of the difference between the total number of zeros of its real and
imaginary parts is exactly 2.
Submitted September 12, 2016. Published December 10, 2016.
Math Subject Classifications: 34C10, 34B25.
Key Words: SturmLiouville; nondefinite; indefinite; spectrum; oscillation;
Dirichlet problem; turning point.
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Mervis Kikonko
Department of Engineering Sciences and Mathematics
Lulea University of Technology
SE971 87 Lule{\aa}, Sweden
email: mervis.kikonko@ltu.se, mervis.kikonko@unza.zm


Angelo B. Mingarelli
School of Mathematics and Statistics
Carleton University
Ottawa, ON, Canada, K1S 5B6
email: angelo@math.carleton.ca

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