Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 10, pp. 1-19.
Title: A second eigenvalue bound for the Dirichlet Schrodinger equation
wtih a radially symmetric potential
Author: Craig Haile (College of the Ozarks, Point Lookout, MO, USA)
Abstract:
We study the time-independent Schrodinger
equation with radially symmetric potential
$k|x|^\alpha$, $k \ge 0$, $k \in \mathbb{R}, \alpha \ge 2$
on a bounded domain $\Omega$ in $\mathbb{R}^n$, $(n \ge 2)$
with Dirichlet boundary conditions.
In particular, we compare the eigenvalue $\lambda_2(\Omega)$ of
the operator $-\Delta + k |x|^\alpha $
on $\Omega$ with the eigenvalue $\lambda_2(S_1)$
of the same operator $-\Delta +kr^\alpha$ on a ball $S_1$,
where $S_1$ has radius such that
the first eigenvalues are the same ($\lambda_1(\Omega) = \lambda_1(S_1)$).
The main result is to show
$\lambda_2(\Omega) \le \lambda_2(S_1)$.
We also give an extension of the main
result to the case of a more general elliptic eigenvalue problem
on a bounded domain $\Omega$ with Dirichlet boundary conditions.
Submitted August 24, 1999. Published January 28, 2000.
Math Subject Classifications: 35J10, 35J15, 35J25, 35P15.
Key Words: Schrodinger eigenvalue equation; Dirichlet boundary conditions;
eigenvalue bounds; radially symmetric potential.