I have this question concerning the Hermite operator on $\mathbb{R}^n$ (defined by $Hf:=-\Delta f+|x|^2f$). We know that $H$ is a densely-defined unbounded operator on $L^2(\mathbb{R}^n)$ and that $\mathcal{E}=\{2k+n\}_{k\in\mathbb{N}_0}$ is the set of its eigenvalues (the corresponding eigenfunctions are the Hermite polynomials). In general, we know that the spectrum $\sigma(T)$ of some operator $T$ contains the set of the eigenvalues $\mathcal{E}_T$ of $T$, but how do I prove that they coincide for $T=H$ (or, there is any theorem that grants it)?
Moreover, how do I prove that the spectral decomposition of $H$ is given by $\sum_k(2k+n)P_k$ where $P_k$ is the projection on the $k$-th eigenspace? I guess I shall prove that the projection valued measure $E$ given by the spectral theorem coincides with the counting measure and, then, invoke the spectral theorem (for unbounded operators).
If you have any good reference, please help me.
Thank you in advance.
As you have mentioned you have an eigenbasis given be the Hermite polynomals. Since every basisvector is only an eigenvalue once, the basis spans the entire Hilbert space and the set of eigenvalues is discrete you obtain that the spectrum is the set of eigenvalues and the spectral decomposition follows.