(Someone may please change the title if they can think of a better one)
We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that
$\int_{-\infty}^{\infty} |\psi(x)|^2 dx \lt \infty $
Now I have to show that there are functions in $\mathcal{H}$ such that $Q\phi(x)=x\psi(x)$ is not in $\mathcal{H}$ . Does it suffice if I think of a counterexample.. but even then, the only two square integrables i can think of are the Gaussian and the dirac-delta "function". For both the cases, the corresponding $x\psi(x)$ is also L2. Could someone give me more examples of functions that are square integrable over the entire real axis?
Furthermore, if I now consider the function space $\Omega$ with the infinite set of conditions,
$\int_{-\infty}^{\infty} |\psi(x)|^2 (1+|x|^n)dx \lt \infty$ for $n=0,1,2,\cdots$
I have to show now, that for any $\psi(x)$ in $\Omega$ the function $Q\phi(x)=x\psi(x)$ is also in $\Omega$. I can think of a weak argument like
$\int_{-\infty}^{\infty} |Q\psi(x)|^2 (1+|x|^n)dx = \int_{-\infty}^{\infty} |\psi(x)|^2 (x^2+|x|^{n+2})dx \approx \int_{-\infty}^{\infty} |\psi(x)|^2 (1+|x|^n)dx \lt \infty $
The last step because we would be concerned only with the behaviour of $x$ when it is very large, as the function would have to go to zero at large $x$ for it to be in $\Omega$.
I am not so sure about this, and face a basic conceptual difficulty. Can we conclude if $\int_{-\infty}^{\infty} |\psi(x)|^2 dx \lt \infty $ then $\psi(x)$ should fall off as $1/x^2$?