Ground-state eigenfunction of a confining Schrödinger operator
= Ground-state eigenfunction of a confining Schrödinger operator
The minimum of
$$
\|\nabla u\|_2^2+\|V^{1/2}u\|_2^2
$$
subject to $\|u\|_2=1$ is attained because $\Sigma_V$ embeds compactly into $L^2$. Replacing a minimizer by its <absolute value function>[absolute value] does not increase its energy. The <Euler-Lagrange equation> therefore produces a nonnegative eigenfunction of $-\Delta+V$ for its lowest eigenvalue.