Electron. J. Diff. Eqns., Vol. 2000(2000), No. 10, pp. 1-19.

A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential

Craig Haile

We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^\alpha$, $k \ge 0$, $\alpha \ge 2$ on a bounded domain $\Omega$ in ${\Bbb R}^n$, $(n \ge 2)$ 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 with Dirichlet boundary conditions.

Submitted August 24, 1999. Published Janaury 28, 2000.
Math Subject Classifications: 35J10, 35J15, 35J25, 35P15.
Key Words: Schrodinger eigenvalue equation, Dirichlet boundary conditions, eigenvalue bounds, radially symmetric potential.

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Craig Haile
Department of Mathematics and Physics
College of the Ozarks
Point Lookout, MO 65726-0017, USA
e-mail: haile@cofo.edu

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