$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ##13.4. About spectrum > 1. [Definitions and classification](#sect-13.4.1) > 2. [Spectrum: examples](#sect-13.4.2) > 3. [Spectrum: explanations](#sect-13.4.3) ###Definitions and classification ------------------ ####Definitions Let $\mathsf{H}$ be a Hilbert space (see [Definition 4.3.3](../S4.3.html#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. 1. 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} 2. 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 1. Operator $L$ is *closed* if $u\_n\to u$, $Lu\_n\to f$ imply that $u\in D(L)$ and $Lu=f$; 2. 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} 3. 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.** 1. For bounded operators "symmetric" equals "self-adjoint"; 2. 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; 3. Self-adjoint operators have many properties which symmetric but not self-adjoint operators do not have; 4. In Quantum Mechanics *observables* are self-adjoin operators. **Theorem 1.** The following statements are equivalent: 1. $L$ is self-adjoint; 2. $L$ 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 $u\in D(L)$; 3. 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 1. $z\in \mathbb{C}$ belongs to the *resolvent set* of $L$ if $(L-z)^{-1}: \mathsf{H}\to D(L)$ exists and is a bounded operator: $(L-z)^{-1}(L-z)=I$, $(L-z)(L-z)^{-1}=I$. 2. $z\in \mathbb{C}$ 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.** 1. Resolvent set is always open (and spectrum is always closed) subset of $\mathbb{C}$; 2. If $L$ is self-adjoint then $\sigma (L)\subset \mathbb{R}$; 3. If $L$ is bounded then $\sigma (L)$ is a bounded set. ####Classification *Not all points of the spectrum are born equal!* From now on we consider only self-adjoint operators. **Definition 5.** 1. $z$ 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! 2. The set of all eigenvalues is called *pure point spectrum*; 3. 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. ###Spectrum: examples **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](../Chaper6/S6.3.html#sect-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](../Chapter8/S8.1.html#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 $E\_n<0$. 1. If $|V(x)|\le M(|x|+1)^{-m}$, $m>2$ the number of eigenvalues is finite. 2. 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*](http://en.wikipedia.org/wiki/Gamma_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](#example-13.4.3) 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](../Chapter4/S4.B.html#definition-4.B.1). Then $L$ has a *band spectrum*. Namely on the *elementary cell* [Definition 4.B.3](../Chapter4/S4.B.html#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{gather} these spectral bands can overlap. The spectrum $\sigma(L)$ is continuos. 1. 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. 2. As dimension $d\ge 2$ only finite number of spectral gaps could be open. 3. 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 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*](http://en.wikipedia.org/wiki/Almost_Mathieu_operator) (which appears in the study of [*quantum Hall effect*](http://en.wikipedia.org/wiki/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 $|\lambda|= 1$. 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*](http://en.wikipedia.org/wiki/Cantor_set). **Remark 4.** 1. [Example 8](#example-13.4.8) was completely investigated only in the end of the 20-th century. 2. [Example 9](#example-13.4.9) was completely investigated only in the 21-st century. ###Spectrum: explanations ####Landau levels Consider [Example 3](#example-13.4.3): 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](./Chapter4/S4.C.html) 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. ####Band spectrum Consider [Example 8](#example-13.4.8): 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. _________ [$\Leftarrow$](./S13.3.html)  [$\Uparrow$](./contents.html)  [$\Rightarrow$](./S13.5.html)