Electron. J. Diff. Equ., Vol. 2010(2010), No. 133, pp. 1-10.

Spectral concentration in Sturm-Liouville equations with large negative potential

Bernard J. Harris, Jeffrey C. Kallenbach

Abstract:
We consider the spectral function, $\rho_{\alpha} (\lambda)$, associated with the linear second-order question
$$
 y'' + (\lambda - q(x)) y = 0 \quad \hbox{in } [0, \infty)
 $$
and the initial condition
$$
 y(0) \cos (\alpha) + y' (0) \sin (\alpha) = 0, \quad
 \alpha \in [0, \pi).
 $$
in the case where $q (x) \to - \infty$ as $x \to \infty$. We obtain a representation of $\rho_0 (\lambda)$ as a convergent series for $\lambda > \Lambda_0$ where $\Lambda_0$ is computable, and a bound for the points of spectral concentration.

Submitted August 27, 2009. Published September 14, 2010.
Math Subject Classifications: 34L05, 34L20.
Key Words: Spectral theory; Schrodinger equation.

Show me the PDF file (219 KB), TEX file, and other files for this article.

Bernard J. Harris
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL 60115, USA
email: harris@math.niu.edu
Jeffrey C. Kallenbach
Department of Mathematical Sciences
Siena Heights University
Adrian, MI 49221, USA
email: jkallenb@siennaheights.edu

Return to the EJDE web page