Density of smooth functions in weighted Sobolev space

Question

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\mathcal{C}^{\infty}_{0}(\mathbb{R})$ is dense in the weighted Sobolev spaces $H^{k}(\mathbb{R},\rho(x)dx)$, for $k\in\mathbb{N}^{*}$. In the article

Zhikov, V. V., Weighted Sobolev spaces, Sb. Math. 189, No. 8, 1139-1170 (1997); translation from Mat. Sb. 189, No. 8, 27-58 (1998). ZBL0919.46026.

which can be found here

the author states that this is always true in the one dimensional case (for $k=1$).. But the proof, which is at page 1156, show the result on the interval $\Omega = [0,1]$ and not on $\mathbb{R}$. Am I missing something? Is the result true or false on $\mathbb{R}$? Is there a better reference for this result? Also, how to show the result for all $k\in\mathbb{N}^{*}$ once the result is proven for $k=1$?

Answer 1

For $k=1$, the proof works the same on $\mathbb{R}$ ; you only need to check that compactly supported functions (no smoothness here) are dense in $H^1(\mathbb{R},\rho(x)dx)$ and this can be done using a sequence of smooth bumped functions converging to $1$. For higher order Sobolev space, I would just iterate the formula (you start to approach the higher derivative in $L^2(\mathbb{R},(1+\rho(x))dx)$ and then deduce the approximation of the less order ones recursively). Of course all this collapse in dimension larger than $1$ !