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.