The operator $L=-\Delta+V$ (where $V(x)$ is the potential) satisfies the maximum principle: if $f\ge0$ and $f\not\equiv0$, then the solution $u$ of $-\Delta u+Vu=f$ exists, is unique and satisfies $u>0$. Then apply the Krein--Rutman Theorem to $L^{-1}$ ; this is the infinite-dimensional version of Perron-Frobenius Theorem, the latter applying to positive matrices. You find that the spectral radius is an eigenvalue, a simple one, associated with a positive eigenfunction.

Say that $V$ be bounded by below. Up to the addition of a large enough constant, you may assume that $V\ge0$. Then argue as follow.

The operator $L=-\Delta+V$ (where $V(x)$ is the potential) satisfies the maximum principle: if $f\ge0$ and $f\not\equiv0$, then the solution $u$ of $-\Delta u+Vu=f$ exists, is unique and satisfies $u>0$. Then apply the Krein--Rutman Theorem to $L^{-1}$ ; this is the infinite-dimensional version of Perron-Frobenius Theorem, the latter applying to positive matrices. You find that the spectral radius is an eigenvalue, a simple one, associated with a positive eigenfunction.