$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$

Let $\mathsf{H}$ be a Hilbert space (see Definition 4.3.3).

**Definition 1.**
Linear operator $L:\mathsf{H}\to\mathsf{H}$ is *bounded* if
\begin{equation}
\| Lu\|\le M\|u\|\qquad \forall u\in \mathsf{H};
\label{eq-13.4.1}
\end{equation}
the smallest constant $M$ for which it holds is called *operator norm* of $L$ and denoted $\|L\|$.

**Definition 2.**
Let $L:\mathsf{H}\to\mathsf{H}$ be a bounded linear operator.

- Adjoint operator $L^*$ is defined as \begin{equation} (Lu, v)= (u,L^*v) \qquad \forall u,v\in \mathsf{H}; \label{eq-13.4.2} \end{equation}
- Operator $L$ is
*self-adjoint*if $L^*=L$.

However one needs to consider also unbounded operators. Such operators not only fail (\ref{eq-13.4.1}) but they are not defined everywhere.

**Definition 3.**
Consider a linear operator $L:D(L)\to \mathsf{H}$ where $D(L)$ is a linear subset in $\mathsf{H}$ (i.e. it is a linear subspace but we do not call it this way because it is not closed) which is *dense* in $\mathsf{H}$ (i.e. for each $u\in \mathsf{H}$ there exists a sequence $u_n \in D(L)$ converging to $u$ in $\mathsf{H}$). Then

- Operator $L$ is
*closed*if $u_n\to u$, $Lu_n\to f$ imply that $u\in D(L)$ and $Lu=f$; - Operator $L$ is
*symmetric*if \begin{equation} (Lu, v)= (u,L v) \qquad \forall u,v\in D(L); \label{eq-13.4.3} \end{equation} - Symmetric operator $L$ is
*self-adjoint*if $(L\pm i)^{-1}:\mathsf{H}\to D(L)$ exist: $(L\pm i)(L\pm i)^{-1}=I$, $(L\pm i)^{-1}(L\pm i)=I$ (identical operator)

**Remark 1.**

- For bounded operators "symmetric" equals "self-adjoint";
- Not so for unbounded operators. F.e. $Lu=-u''$ on $(0,l)$ with $D(L)=\{u(0)=u'(0)=u(l)=u'(l)=0\}$ is symmetric but not self-adjoint;
- Self-adjoint operators have many properties which symmetric but not self-adjoint operators do not have;
- In Quantum Mechanics
*observables*are self-adjoin operators.

**Theorem 1.**
The following statements are equivalent:

- $L$ is self-adjoint;
- $L$ generates unitary group $e^{itL}$ ($t\in \mathbb{R}$: $\| e^{itL} u\|=\|u\|$, $e^{i(t_1+t_2)L}= e^{it_1L}e^{it_2L}$, $u\in D(L)\implies e^{itL}u\in D(L)$, $\frac{d\ }{dt}e^{itL} u= L e^{itL}u$ for all $u\in D(L)$ (and conversely, if $e^{itL}u$ is differentiable by $t$ then $u\in D(L)$;
- Exist spectral projectors -- operators $\theta (\tau -L)$ ($\theta(\tau)=0$ as $\tau\le 0$ and $\theta(\tau)=1$ as $\tau>0$) such that $\theta(\tau -L)$ are orthogonal projectors, $\theta (\tau_1-L)\theta (\tau_2-L)=\theta (\tau-L)$ with $\tau=\min (\tau_1,\tau_2)$, $\theta (\tau-L)u\to 0$ as $\tau\to -\infty$; $\theta (\tau-L)u\to u$ as $\tau\to +\infty$; $\theta (\tau-L)u\to \theta (\tau^*-L)$ as $\tau\to \tau^*-0$ and \begin{equation} f(L)=\int f(\tau)d_\tau \theta (\tau-L) \label{eq-13.4.4} \end{equation}

**Definition 4.**
Let us consider operator $L$ (bounded or unbounded). Then

- $z\in \mathbb{C}$ belongs to the
*resolvent set*of $L$ if $(L-z)^{-1}: \mathsf{H}\to D(L)$ exists and is a bounded operator: $(L-z)^{-1}(L-z)=I$, $(L-z)(L-z)^{-1}=I$. - $z\in \mathbb{C}$ belongs to the
*spectrum*of $L$ if it does not belong to its resolvent set. We denote spectrum of $L$ as $\sigma (L)$.

**Remark 2.**

- Resolvent set is always open (and spectrum is always closed) subset of $\mathbb{C}$;
- If $L$ is self-adjoint then $\sigma (L)\subset \mathbb{R}$;
- If $L$ is bounded then $\sigma (L)$ is a bounded set.

*Not all points of the spectrum are born equal!* From now on we consider only self-adjoint operators.

**Definition 5.**

- $z$ is an
*eigenvalue*if there exists $u\ne 0$ s.t. $(A-z)u=0$. Then $u$ is called*eigenvector*(or*eigenfunction*depending on context) and $\{u:\, (A-z)u=0\}$ is an*eigenspace*(corresponding to eigenvalue $z$). The dimension of the eigenspace is called a*multiplicity*of $z$. Eigenvalues of multiplicity $1$ are*simple*, eigenvalues of multiplicity $2$ are*double*, ... but there could be eignvalues of infinite multiplicity! - The set of all eigenvalues is called
*pure point spectrum*; - Eigenvalues of the finite multiplicity which are isolated from the rest of the spectrum form a
*discrete spectrum*; the rest of the spectrum is called*essential spectrum*.

**Definition 6.**
$z\in \mathbb{C}$ belongs to *continuous spectrum* if $z$ is not an eigenvalue and inverse operator $(L-z)^{-1}$ exists but is not a bounded operator (so its domain $D((L-z)^{-1}$ is dense).

**Remark 3.**
Continuous spectrum could be classified as well. The difference between *absolutely continuous* and *singular continuous* spectra will be illustrated but one can define also multiplicity of continuous spectrum as well. However one needs a *Spectral Theorem* to deal with these issues properly.

**Example 1.**
Schrödinger operator
\begin{equation}
L=-\frac{1}{2}\Delta + V(x)
\label{eq-13.4.5}
\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 6.3.2.
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 Definition 8.1.1.

**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{eq-13.4.6}
\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{eq-13.4.5}) 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 $E_n<0$.

- If $|V(x)|\le M(|x|+1)^{-m}$, $m>2$ the number of eigenvalues is finite.
- For
*Coulomb potential*$V(x)=-Z|x|^{-1}$ ($Z>0$) $E_n=-\frac{Z^2}{4n^2}$ of multiplicity $n^2$, $n=1,2,\ldots$.

**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{eq-13.4.7}
\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{eq-13.4.8}
\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{eq-13.4.9}
\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{eq-13.4.5}) with periodic potential in $\mathbb{R}^d$: $V(x+a)=V(x)$ for all $a\in \Gamma$ where $\Gamma$ is a *lattice of periods*, see Definition 4.B.1. Then $L$ has a *band spectrum*.

Namely on the *elementary cell* Definition 4.B.3 $\Omega$ 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*
\begin{gather}
\sigma_n:=[\min _{k\in \Omega^*} E_n(k) ,\max _{k\in \Omega^*} E_n(k)]: \label{eq-13.4.10}\\
\sigma(L) =\bigcup_{n=1}^\infty \sigma_n;
\label{eq-13.4.11}
\end{equation}
these spectral bands can overlap. The spectrum $\sigma(L)$ is continuos.

- As dimension $d=1$ we can do better than this: $E_n(k)$ are increasing (decreasing) functions of $k$ on $(0,\pi/a)$ (where $a$ is the period) as $n$ is odd (respectively even) and \begin{equation}E_n^*:=\max _{k\in [0,\pi/a]} E_n(k)\le E_{(n+1)*}:=\min _{k\in [0,\pi/a]} E_{n+1}(k)\label{eq-13.4.12}\end{equation} and for
*generic potential*$V(x)$ all inequalities are strict and all*all spectral gaps*$(E_n^*,E_{(n+1)*})$ are open. - As dimension $d\ge 2$ only finite number of spectral gaps could be open.
- Perturbation of such operator $L$ by another potential $W(x)$, $W(x)\to 0$ as $|x|\to \infty$ could can add a finite or infinite number of eigenvalues in
*spectral gaps*. They can accumulate only to the borders of the spectral gaps.

**Example 8.**
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{eq-13.4.13}
\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*.

**Remark 4.**

- Example 8 was completely investigated only in the end of the 20-th century.
- Example 9 was completely investigated only in the 21-st century.

Consider 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{eq-13.4.14}
\end{equation}
with $B>0$ (or $B<0$) has a pure point spectrum. For simplicity assume that $B>0$. We apply a *gauge transformation* which for Schrödinger operator means multiplying it from the left and right by $e^{i\hbar^{-1} \phi (x)}$ and $e^{-i\hbar^{-1} \phi (x)}$ respectively with a real-valued $\phi$ (which is an unitary transformation) and replaces $-i\hbar\nabla$ by $-i\hbar\nabla - (\nabla\phi)$ (which is equivalent to changing vector potential $\mathbf{A}(x)$ by $\nabla \phi$ which in turn does not change $\nabla \times \mathbf{A}$. Taking $\hbar=1$ and $\phi= \frac{1}{2}B xy$ we arrive to
\begin{equation*}
L'=\frac{1}{2} (-i\partial_x -B y)^2 +
\frac{1}{2} (-i\partial_y )^2;
\end{equation*}
then making Fourier transform by $x\mapsto \xi$ we get
\begin{equation*}
L''=\frac{1}{2} (-\xi -B y)^2 +
\frac{1}{2} (-i\partial_y )^2;
\end{equation*}
and plugging $y=B^{-\frac{1}{2}} (y_{\textsf{new}} -B^{-1}\xi)$ we get
\begin{equation*}
\frac{1}{2} B (-\partial_y^2 +y^2)
\end{equation*}
which is a harmonic oscillator multiplied by $B$ and in virtue of Section 4.C its spectrum consists of eigenvalues
$E_n = |B|(n+\frac{1}{2})$, $n=0,1,2,\ldots$ which are called *Landau levels.*

However there is a "hidden variable" $\xi$, so eigenfunctions Hermite functions of $y$ but multiplied by arbitrary functions $C(\xi)$ rather than by constants which implies that these eigenvalues have constant multiplicities.

Consider Example 8: Schrödinger operator with periodic potential in $\mathbb{R}^d$: $V(x+a)=V(x)$ for all $a\in \Gamma$ where $\Gamma$ is a lattice of periods.

Let us decompose function $u(\mathbf{x})$ into $n$-dimensional Fourier integral \begin{equation*} u(x)= \iiint _{\mathbb{R}^n} e^{i \mathbf{k}\cdot \mathbf{x}} \hat{u}(\mathbf{k})\,d^n\mathbf{k}, \end{equation*} then replace this integral by a sum of integrals over dual elementary cell $\Omega^*$ shifted by $\mathbf{n}\in \Gamma^*$ \begin{equation*} \sum_{ \mathbf{n}\in \Gamma^* }\iiint _{\Omega^* +\mathbf{n}} e^{i \mathbf{k}\cdot \mathbf{x}} \hat{u}(\mathbf{k})\,d^n\mathbf{k}, \end{equation*} then change variable $\mathbf{k}=\mathbf{k}_{\textsf{new}}+\mathbf{n}$ \begin{equation*} \iiint _{\Omega^* } e^{i (\mathbf{k}+\mathbf{n})\cdot \mathbf{x}} \Bigl( \underbrace{\sum_{ \mathbf{n}\in \Gamma^* } e^{i \mathbf{n}\cdot \mathbf{x}} \hat{u}(\mathbf{k}+\mathbf{n})}_{=U(\mathbf{k},\mathbf{x})}\Bigr)d^n\mathbf{k}, \end{equation*} we observe that $U(\mathbf{k},\mathbf{x})$ is quasiperiodic with quasimomentum $\mathbf{k}$.

In advanced Real Analysis it would be a decomposition of our Hilbert space $\mathsf{H}=L^2(\mathbb{R}^n)$ into *direct integral of Hilbert spaces* $\mathsf{H}(\mathbf{k})$ of such functions, and our operator is acting in each of those spaces separately, with a spectrum $\sigma (L(\mathbf{k}))=\{E_n (\mathbf{k}): n=1,2,\ldots\}$. This implies that $L$ has a *band spectrum*: it 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;
\end{equation*}
these spectral bands can overlap. On can prove that $E_n (\mathbf{k})$ really depend on $\mathbf{k}$ and are not taking the same value on some set of *non–zero measure* (another notion from Real Analysis) which implies that the spectrum $\sigma(L)$ is continuos.