**MSC:**- 35B50 Maximum principles
- 35J10 Schrodinger operator, See also {35Pxx}
- 35P30 Nonlinear eigenvalue problems, nonlinear spectral theory for PDO
- 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

u=f(x) $ in $L ^2(\mathbb{R} ^N)$ is extended to potentials $q$ non

necessarily radial. The anti-maximum is proved in the following form:

Let $\varphi_1$ denote the positive eigenfunction associated with the

principal eigenvalue $\lambda _1$ of the Schr\"odinger operator ${\cal

A}=-\Delta + q(x)\bullet$ in $L ^2(\mathbb{R} ^N)$. Assume the potential

$q(x)$ grows fast enough near infinity, and the function $f$ satisfy

$f\not\equiv 0$ and $0\leq f/\varphi_1\leq C\equiv const$ a.e. in $\mathbb{R}

^N$.

Then there exists a positive number $\delta$

(depending upon $f$)

such that, for every

$\lambda\in (\lambda_1, \lambda_1 + \delta)$,

the inequality

$ u\leq -c\varphi_1$

holds a.e.\ in $\mathbb{R} ^N$,

where $c$ is a positive constant depending upon $f$ and $\lambda$.