Let $\alpha \in \mathbb{C}$, with $\Re(\alpha) , \Im(\alpha)>0$ and $H$ the operator defined in $L^2(\mathbb{R})$ by
$$H(u):= -\frac{d^2}{dx^2}u +\alpha x^2u.$$
How can I show that the spectrum of $H$ coincides with its point spectrum?
Thanks in advance.