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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,      axb,

 
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).

 

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