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 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
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
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 non-real eigenfunctions. Furthermore,
we show that when a non-real 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: Sturm-Liouville; non-definite; indefinite; spectrum; oscillation;
Dirichlet problem; turning point.
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Mervis Kikonko Department of Engineering Sciences and Mathematics Lulea University of Technology SE-971 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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