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Volume 12, Number 3, Fall 2004
HIGHER ORDER ASYMPTOTICS OF THE
EIGENVALUES OF STURM-LIOUVILLE
PROBLEMS WITH A TURNING POINT OF
ARBITRARY ORDER
A. JODAYREE AKBARFAM AND ANGELO B. MINGARELLI
Abstract. We consider the differential equation
y" + ( λr(x) - q(x))y = 0, a ≤ x ≤ b,
on the interval [a, b], where [a, b] contains one zero of r(x), the
so called turning point, λ is a real parameter, q, r : [a, b] → R are
continuous. Using a technique used previously by the authors
in [7], we derive the higher-order asymptotic distribution of the
positive and negative eigenvalues associated with this equation
for the Dirichlet problem (i.e., y(a) = 0 = y(b)), on the assumption
that the turning point is of arbitrary order. In particular,
we continue the study of the higher-order asymptotic distribution
of the eigenvalues by focusing on the important special
case where one of the end points a or b is a zero of the weight
function r(x).
(Subscribers Only)
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© 2005, Canadian Applied Mathematics Quarterly (CAMQ) |
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