Asymptotics of Eigenvalues for Sturm-Liouville Problems with an Interior Singularity

✍ Scribed by B.J. Harris; D. Race

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 718 KB
• Volume: 116
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1995.1030

✦ Synopsis

We consider the asymptotic form of the eigenvalues of the linear differential equation

[
-y^{\prime \prime}(x)+q(x) y(x)=\lambda y(x), \quad-\infty<a<x<b<x,
]

where (a<0<b, q(x)) is singular at (x=0), and (y) satisfies appropriate conditions at (a, 0), and (b). This extends previous work of Atkinson and of Harris. In particular, when (q(x)=x^{-k}), Atkinson derived asymptotic formulae which cover the case (1 \leqslant K<\frac{4}{3} ;) Harris's results cover the cases (1 \leqslant K<\frac{3}{2}). We now cover all of the cases (1 \leqslant K<2). Since the methods employed by both of these authors and ourselves apply only to limit circle, non-oscillatory expressions, our results now seem to take problems of this type to their conclusion. 1995 Academic Press, Inc.