= Confining-potential energy space
{title2=$\Sigma_V$}
Let $V$ be a nonnegative, locally bounded <measurable function> on $\mathbb R^d$ with $V(x)\to\infty$ as $|x|\to\infty$. Its confining-potential energy space is
$$
\Sigma_V=\{u\in H^1(\mathbb R^d):V^{1/2}u\in L^2\},\qquad
\|u\|_{\Sigma_V}^2=\|\nabla u\|_2^2+\|V^{1/2}u\|_2^2.
$$
The displayed quantity is a <norm> even though it omits an explicit <L2 norm> term. Choose a ball outside which $V\ge1$. The weighted term controls $\|u\|_2$ outside that ball, and a local <Poincaré inequality> anchored on a surrounding annulus controls the interior <L2 norm> by the gradient and that annular mass. Thus the displayed <norm> is equivalent to the full graph norm containing $\|u\|_2$.
A Cauchy sequence converges in $H^1$ and its weighted values converge in <L2 space>; an almost-everywhere convergent subsequence identifies the latter limit with $V^{1/2}u$. Hence $\Sigma_V$ is a <Hilbert space>. Growth of the potential makes bounded-energy mass tails small, which yields <compact embedding of a confining-potential energy space>.
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