I once read that the first eigenvalue of a Schrödinger operator always is simple, together with an easy proof of it. But I cannot remember where. Does anybody know a reference?
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 KreinRutman Theorem to $L^{1}$ ; this is the infinitedimensional version of PerronFrobenius Theorem, the latter applying to positive matrices. You find that the spectral radius is an eigenvalue, a simple one, associated with a positive eigenfunction. 


Barry Simon's book, "Functional Integration and Quantum Physics", should fit the bill. 

