Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 133, pp. 1-10.
Title: Spectral concentration in Sturm-Liouville equations with large
negative potential
Authors: Bernard J. Harris (Northern Illinois Univ., DeKalb, IL, USA)
Jeffrey C. Kallenbach (Siena Heights Univ., Adrian, MI, USA)
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.