Example of self-adjoint linear operator with pure point spectrum on an infinite-dimensional separable Hilbert space Let $\mathcal{H}$ be an infinite-dimensional, complex, separable Hilbert space.
Besides the well-known one-dimensional Harmonic oscillator on $\mathcal{H}=(\mathcal{L}^{2}(\mathbb{R})\,;d\mathit{l})$, does anyone know some explicit examples (domain, eigenvalues and eigenvectors) of a possibly unbounded self-adjoint linear operator having only a pure-point spectrum with no degenerancy?
References are appreciated.
Thanks in advance.
ADDENDUM
In view of TrialAndError's reply, I would like to add that an example on a functional space (e.g., $\mathcal{L}^{2}(M)$ with $M$ a measure space$) would be better.

Canonical spaces...
Given $\Lambda\subseteq\mathbb{C}$ and $\ell^{2}(\overline{\Lambda})$.
Denote for readability:
$$A\in\mathcal{B}(\mathbb{C}):\quad1_A(\lambda):=\chi_A(\lambda)$$
Construct normal operator:*
$$E(A)\varphi:=1_{\overline{\Delta}\cap A}\varphi:\quad N:=\int\lambda\mathrm{d}E(\lambda)$$
Then it has spectrum:**
$$\sigma(N)=\sigma_0(N)=\overline{\Lambda}\subseteq\mathbb{C}$$
Note that the set was arbitrary!!
*See the thread: Constructions
**See the thread: Special Spectrum
