The question:
Consider the function space $\Omega$ which consists of all $\phi(x)$ that satisfy the infinite set of conditions, $$\int_{-\infty}^{\infty}|\phi(x)|^2(1+|x|^n)<\infty$$ for any $n\in \mathbb{N}$. Show that for any $\phi(x) \in \Omega$ the function $Q\phi(x) = x\phi(x)$ is also in $\Omega$. (These results are expressed by the statement that the domain of the operator $Q$ includes all of $\Omega$, but not all of $\mathscr{H}$ (the corresponding Hilbert space).
My attempt at a proof is as follows:
Suppose $\phi \in \Omega$. Now we have for any $n\in \mathbb{N}$ that $$ \int_{-\infty}^{\infty}|Q\phi(x)|^2(1+|x|^n) = \int_{-\infty}^{\infty}|x\phi(x)|^2(1+|x|^n) = \int_{-\infty}^{\infty}|\phi(x)|^2(|x|^2+|x|^{n+2})$$ $$\leq \int_{-1}^{1}|\phi(x)|^2(1+|x|^{n+2}) + 2\int_{1 }^{\infty}|\phi(x)|^2(|x|^2+|x|^{n+2})$$ but I can't figure out how to bound the second term (the first term is finite since the corresponding integral over all the real line is finite and since the integrand is everywhere nonnegative).
Can you help me finish the proof (this is not homework, this is self-study from Ballentine's quantum mechanics text as should be clear from my post history)?
You're almost there! For the last part, note that $|x|^2+|x|^{n+2}\le |x|^{n+2}+|x|^{n+2}=2|x|^{n+2}\le2(1+|x|^{n+2})$ when $|x|\ge1$.