I need to find the weak formulation of this equation on $ H^1_W(\mathbb{R}^n) $ the weighted Sobolev space.
$$ \left(\dfrac{-1}{2m}\Delta + V(x) - \lambda\right)u = f $$
With $V(x)$ bounded from $\mathbb{R}^n$ to $\mathbb{R}$ and $f \in L^2(\mathbb{R}^n)$.
I want to multiply the equation by a test function $ v \in H^1_W(\mathbb{R}^n) $ and integrate by parts using Green's identity but as far as I know I can only use the latter on a bounded region of $ \mathbb{R}^n $. Any indications ? Thanks.
Suppose $w \in A_p$ is a "nice" weight function. Then, you can find a sequence $C_c^{\infty}(\mathbb{R}^n)$ such that $$ v_n \to v $$ in $H^1_w(\mathbb{R}^n)$. You can find this result for example here: https://www.jams.jp/scm/contents/Vol-10-1/10-6.pdf. But then it is just standard procedure: $$ \int_{\mathbb{R}^n} v_n \Delta u \; dx=-\int_{\mathbb{R}^n} \nabla v_n \cdot \nabla u \; dx. $$ To conclude and pass to the limit on both sides, you need to have a condition on your weight function, which ensures, for example, that $H^1_w(\mathbb{R}^n) \subset H^1(\mathbb{R}^n)$ is a continuous embedding. You can finde maybe inspiration here: https://www.acadsci.fi/mathematica/Vol19/kilpelai.pdf.
I should mention that weighted sobolev spaces usually arise in the context of an elliptic operator of the form $$ -\nabla \cdot (w(x)A(x)\nabla u(x)), $$ however, in your case, you would just end up with $A(x)$ to be the identity and more importantly $w(x)=\frac{1}{2m}$ which is constant. The norm on your weighted Sobolev space is equivalent to the standard norm and hence doesnt make much of a difference. With more information, I could probably give a better answer as well.