Consider Hermite operator (harmonic oscillator) $H=-\Delta+|x|^2.$ It is known that hermite functions are eigen functions for $H$: $$H(\Phi_\alpha)= (2|\alpha| +d) \Phi_{\alpha}, \quad (|\alpha |= \alpha_{1}+ ...+ \alpha_d, \alpha_{i}\in \mathbb N).$$ $( \Phi_\alpha(x) = \Pi_{j=1}^d h_{\alpha_j}(x_j), \quad h_k(x) = (\sqrt{\pi}2^k k!)^{-1/2} (-1)^k e^{\frac{1}{2}x^2} \frac{d^k}{dx^k} e^{-x^2}), x\in \mathbb R^d$
Also it known that $\{\Phi_{\alpha}\}$ forms ONB for $L^2(\mathbb R^d): f= \sum_{\alpha \in \mathbb N^d} \langle f, \Phi_{\alpha} \rangle \Phi_{\alpha}.$
My Questions are following: (1) What is the domain of harmonic oscillator $H$? Can say it is $L^2(\mathbb R^d)$? (2) Can we say that $H$ is self-adjoint operator $L^2(\mathbb R^d)$? (3) Can we expect to use spectral theorem to get the spectral decomposition for $H$, that $$Hf=\sum_{\alpha \in \mathbb N^d} (2|\alpha| +d) \langle f, \Phi_{\alpha} \rangle \Phi_{\alpha} = \sum_{k=0}^\infty (2k+d)P_kf$$ where $P_kf(x) = \sum_{|\alpha|=k} \langle f,\Phi_\alpha\rangle \Phi_\alpha.$
My attempt: "Spectral Theorem: Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal H$. Then there is a unique projection-valued measure $\mu^A$ on the spectrum $\sigma(A)\subseteq\mathbb R$ with values in the bounded operators $\mathcal B(\mathcal H)$ such that $$ \int_{\sigma(A)}\lambda\,d\mu^A(\lambda)=A. $$"
(i)We may wish to apply spectral theorem for $H$. (ii)I think the spectrum of $H$, that is, $\sigma(H)=\{ 2|\alpha|+d: \alpha \in \mathbb N^d \}\subset \mathbb N$ (How should I justify this)
(please correct me if I'm wrong!!) (iii) I wish to find spectral valued measure $\mu^H$ on $\sigma (H)$ (iv) How to interpret: $\int_{\sigma (H) } \lambda \mu^H(\lambda)$