Weak lower semicontinuity of a bounded-domain Schrodinger energy

On bounded $\Omega$ with $V\in L^{\infty }(\Omega )$, weak convergence in $H_0^1(\Omega )$ makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term $\int V{u}^2$ continuous. Thusis weakly lower semicontinuous.