Geodesic
A second eigenvalue bound for the Dirichlet-Schrödinger equation with a radially symmetric potential
Electronic Journal of Differential Equations, Tome 2000 (2000).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35J10, 35J15, 35J25, 35P15
Keywords: Schrödinger eigenvalue equation, Dirichlet boundary conditions, eigenvalue bounds, radially symmetric potential 
@article{EJDE_2000__2000__a73,
     author = {Haile, Craig},
     title = {A second eigenvalue bound for the {Dirichlet-Schr\"odinger} equation with a radially symmetric potential},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2000},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a73/}
}
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Haile, Craig. A second eigenvalue bound for the Dirichlet-Schrödinger equation with a radially symmetric potential. Electronic Journal of Differential Equations, Tome 2000 (2000). http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a73/