Confining-potential energy space
= Confining-potential energy space
{title2=$\Sigma_V$}
For a nonnegative potential $V(x)\to\infty$ as $|x|\to\infty$, the confining-potential energy space is
$$
\Sigma_V=\{u:\nabla u\in L^2,\ V^{1/2}u\in L^2\},
\qquad
\|u\|_{\Sigma_V}^2=\|\nabla u\|_2^2+\|V^{1/2}u\|_2^2.
$$
A local <Poincare inequality> controls the missing $L^2$ term, so this energy norm makes $\Sigma_V$ a <Hilbert space>.