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.

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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

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