





Volume 12, Number 3, Fall 2004
HIGHER ORDER ASYMPTOTICS OF THE
EIGENVALUES OF STURMLIOUVILLE
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 higherorder 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 higherorder 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).
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