Infinite energy states for an harmonic oscillator?

[Catria]

So, I've read conference proceedings and they appear to talk about counter-intuitive it was to create an infinite-energy state for the harmonic oscillator with a normalizable wave function (i.e. a linear combination of eigenstates). How exactly could those even exist in the first place?

 

[rubi]

Try $\Psi(x) = \frac{1}{1+x^2}$. It's clearly square-integrable, but if you apply the harmonic oscilllator Hamiltonian, you get a function that converges to a non-zero constant for $x\rightarrow\infty$. The reason for why there can be such states is that the Hamiltonian of the harmonic oscillator is unbounded. The $\Psi$ I mentioned doesn't lie in the domain of $\hat H$. Most physical observables correspond to unbounded operators.