$\newcommand{\erf}{\operatorname{erf}}$
$\newcommand{\const}{\mathrm{const}}$
This is advanced material. Audience: mathematics, theoretical physics. We consider different operators appearing in different mathematical models in physics. For definitions see Appendix L.
Example 1. Schrödinger operator \begin{equation} L=-\frac{1}{2}\Delta + V(x) \label{equ-M.1} \end{equation} with potential $V(x)\to +\infty$ as $|x|\to \infty$ has a discrete spectrum: its eignevalues $E_n\to +\infty$ have finite multiplicities. In dimension $d=1$ all these eigenvalues are simple, not necessarily so as $d\ge 2$.
Example 2. Consider Laplacian on 2-dimensional sphere which appears after separation of variables for Laplacian in $\mathbb{R}^3$ in spherical coordinates in Subsection 20.2. Then $-\Delta$ has a spectrum $\{E_n= n(n+1): n=0,1,\ldots\}$; $E_n$ is an eigenvalue of multiplicity $(2n+1)$. Corresponding eigenfunctions are spherical harmonics. See Subsection 28.1.
Example 3. Schrödinger operator in 2D with a constant magnetic and no electric field \begin{equation} L=\frac{1}{2} (-i\partial_x -\frac{1}{2}B y)^2 + \frac{1}{2} (-i\partial_y +\frac{1}{2}B y)^2 \label{equ-M.2} \end{equation} with $B>0$ (or $B<0$) has a pure point spectrum. Eigenvalues $E_n = |B|(n+\frac{1}{2})$, $n=0,1,2,\ldots$ have infinite multiplicity and are called Landau levels.
Example 4. "Free" particle Schrödinger operator $L=-\frac{1}{2}\Delta$ in $\mathbb{R}^d$ has a continuous spectrum $[0,+\infty)$.
Example 5. Schrödinger operator (\ref{equ-M.1}) with potential $V(x)\to 0$ as $|x|\to \infty$ has a continuous spectrum $[0,+\infty)$ but it can have a finite or infinite number of negative eignvalues $E_n<0$.
Example 6. Free particle Dirac operator \begin{equation} L=\sum _{j=1}^3 \gamma^j (-i\partial_{x_j}) + \gamma^0 m, \qquad m>0 \label{equ-M.3} \end{equation} (where $\gamma^j$ are Dirac matrices has a continuous spectrum $(-\infty,-m]\cup [m,\infty)$.
Perturbing it by a potential $V(x)$, $V(x)\to 0$ as $|x|\to \infty$ \begin{equation} L=\sum _{j=1}^3 \gamma^j (-i\partial_{x_j}) + m\gamma^0 +V(x) I, \qquad m>0 \label{equ-M.4} \end{equation} can add a finite or infinite number of eigenvalues in spectral gap $(-m,m)$. They can accumulate only to the borders of the spectral gap.
Example 7. Perturbing Example 3 by a potential $V(x)$, $V(x)\to 0$ as $|x|\to \infty$ \begin{equation} L=\frac{1}{2} (-i\partial_x -\frac{1}{2}B y)^2 + \frac{1}{2} (-i\partial_y +\frac{1}{2}B y)^2+V(x) \label{equ-M.5} \end{equation} breaks Landau levels into sequences of eigenvalues $E_{n,k}$, $n=0,1,\ldots$, $k=1,2,\ldots$, $E_{n,k}\to E_n= |B|(n+\frac{1}{2})$ as $k\to \infty$.
Example 8. Consider Schrödinger operator (\ref{equ-M.1}) with periodic potential in $\mathbb{R}^d$: $V(x+a)=V(x)$ for all $a\in \Gamma$ where $\Gamma$ is a lattice of periods. Then $L$ has a band spectrum.
Namely on the elementary cell $\Omega$ consider operator $L(k)$ where $k\in \Omega^*$ is a quasimomentum; $L(k)$ is given by the same formula as $L$ but s defined on functions which are quasiperiodic with quasimomentum $k$. Its spectrum is discrete: $\sigma (L(k))=\{E_n (k): n=1,2,\ldots\}$.
Then spectrum $\sigma (L)$ consists of spectral bands $\sigma_n:=[\min _{k\in \Omega^*} E_n(k) ,\max _{k\in \Omega^*} E_n(k)]$: \begin{equation} \sigma(L) =\bigcup_{n=1}^\infty \sigma_n; \label{equ-M.6} \end{equation} these spectral bands can overlap. The spectrum $\sigma(L)$ is continuos.
Example 9. In the space $\ell^2(\mathbb{Z})$ (which is the space of sequences $u_n$, $n=\ldots, -2,-1,0, 1,2,\ldots$ such that $\|u\|^2:=\sum_{n=-\infty} ^{\infty}|u_n|^2<\infty$) consider almost Mathieu operator (which appears in the study of quantum Hall effect). \begin{equation} (Lu)_n =u_{n+1}+u_{n-1}+2\lambda \cos (2\pi (\theta +n\alpha)) \label{equ-M.8} \end{equation} with $|\lambda|\le 1$. Assume that $\alpha$ is a Diophantine number (which means it is an irrational number which cannot be approximated well by rational numbers; almost all irrational numbers (including all algebraic like $\sqrt{2}$) are Diophantine).
Then the spectrum $\sigma(L)$ is continuous (no eigenvalues!) but it is singular continuous: for any $\varepsilon>0$ it can be covered by the infinite sequence of segments of the total length $<\varepsilon$. As an example of such set see Cantor set.