Compact embedding in $L^2((0,\infty),rdr)$

Question

Let $H$ be the closure of $C^{\infty}_0(0,\infty)$ with respect to the inner product $$(f,g)=\int_0^{\infty}\left(f'(r)g'(r)+h(r)f(r)g(r)\right)rdr,$$ and induced norm $||f||_H^2=(f,f)$, and where $h(r)=1/r^2+\lambda$ with $\lambda>0$. Is this space compactly embedded in $L^2((0,\infty),rdr)$?

Here is what I am able to show:

1. $H$ is continuously embedded in $L^2((0,\infty),rdr)$ and in $W^{1,2}((0,\infty),rdr)$, i.e., \begin{align} \int_0^{\infty}f^2(r)rdr\leq\int_0^{\infty}\left(f_r^2(r)+f^2(r)\right)rdr\leq \max\left\{1,\dfrac{1}{\lambda}\right\}||f||_H^2. \end{align}
2. $H$ is embedded in $W_{loc}^{1,2}(0,\infty)$. So $f$ is continuous on $(0,\infty)$.
3. Every $f\in H$ satisfies $f(0)=0$ and $f(\infty)=0$.
4. Using $f(0)=0$, we have that $H$ is embedded in $L^{\infty}[0,\infty)$.