\usemodule[bib]

\def\mathbb#1{{\blackboard #1}}
\def\pauli{\mathfrak{P}}
\def\mathfrak#1{{\fraktur #1}}


% Matriser av olika typ.
% Paranthesis
\definemathmatrix
[pmatrix]
[left={\left(\,},right={\,\right)}]
% Brackets
\definemathmatrix
[bmatrix]
[left={\left[\,},right={\,\right]}]
% Curly braces
\definemathmatrix
[Bmatrix]
[left={\left\{\,},right={\,\right\}}]
% vertical bars
\definemathmatrix
[vmatrix]
[left={\left\vert\,},right={\,\right\vert}]
% double vertical bars
\definemathmatrix
[Vmatrix]
[left={\left\Vert\,},right={\,\right\Vert}]

\def\PD#1#2{\frac{\partial #1}{\partial #2}}
\def\ann{\mathscr{Q}^{\vphantom{*}}}
\def\mathscr#1{{\gothic #1}}
\def\cre{\mathscr{Q}^*}
\def\dirac{{\mathfrak{D}}}
\def\pform{\mathfrak{p}}
\def\ed{\mfunction{\,d}}
\definemathcommand [dom] [nolop] {\mfunction{Dom}}
\def\bz{\bar{z}}
\def\psip{{\psi_+}}
\def\psim{{\psi_-}}

% \eqref
\definereferenceformat[eqref][left=(,right=)]

\defineenumeration[problem]
        [text=Problem,
        location=serried,
        width=fit,
        indenting=first,
        distance=0.5em,
	way=bysection,
        ]

\subsubsection[ksec:paulitvad]{The Pauli operator in $\mathbb{R}^2$}

A charged spin $1/2$ particle is described by the Pauli Hamiltonian, which acts in $L_2(\mathbb{R}^2)\otimes \mathbb{C}^2$, and is formally defined as
\placeformula[keq:pauli]
\startformula
\pauli = 
\startpmatrix
\NC H-\frac{g}{2}B \NC 0 \NR
\NC 0 \NC H+\frac{g}{2}B\NR
\stoppmatrix.
\stopformula
Here $H$ is the two-dimensional Schrödinger Hamiltonian $H=(-i\nabla -\vec{a})^2$, $B$ is the magnetic field (In two dimensions we identify the two-form and the coefficient function), and $g$ is the {\em gyromagnetic ratio}. We identify the real point $(x^1,x^2)$ with the complex number $z=x^1+ix^2$, and denote a scalar potential of $B$ by $W$,
\startformula
-\Delta W = B.
\stopformula
We set $\Pi_j = -i\PD{}{x^j}-a_j$ and
\startformula
 \ann = \Pi_1 - i \Pi_2,\quad \cre = \Pi_1 + i\Pi_2,
\stopformula
and note that
\placeformula[keq:komm]
\startformula
\ann\cre=\cre\ann+2B,\quad H=\cre\ann+B=\ann\cre-B.
\stopformula
From~\eqref[keq:pauli] and~\eqref[keq:komm] we get
\placeformula[keq:paulig]
\startformula
\pauli = 
\startpmatrix
\NC \cre\ann-\frac{g-2}{2}B \NC 0 \NR
\NC 0 \NC \ann\cre+\frac{g-2}{2}B\NR
\stoppmatrix.
\stopformula
The number $\frac{g-2}{2}$ is called the {\em anomaly factor} of the magnetic moment. Experiments give an anomaly factor of $0.00159$ for the electron~\cite[bovo]. We assume that $g=2$, which is the simplest case. Thus, the Pauli Hamiltonian we study in this thesis is formally defined by
\placeformula[keq:pauliw]
\startformula
\pauli = 
\startpmatrix
\NC \cre\ann \NC 0 \NR
\NC 0 \NC \ann\cre\NR
\stoppmatrix.
\stopformula
The Pauli operator can be written as the square of the Dirac operator
\placeformula[keq:paulia]
\startformula
\pauli = \dirac^2 = \Bigl(\sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\big)\Bigr)^2 = 
\startpmatrix
\NC 0 \NC \cre \NR
\NC \ann \NC 0\NR
\stoppmatrix^2
\stopformula
from which it follows that it is a non-negative operator. Now let us be more precise about the domains. As in the case of the Schrödinger Hamiltonian there is a problem in defining the Pauli Hamiltonian if the magnetic field is too singular. The quadratic form corresponding to $\pauli$ is given by
\placeformula[keq:pform]
\startformula
\pform(\psi,\psi)= \int_{\mathbb{R}^2} \Big|\sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\big)\psi\Big|^2\ed m(x).
\stopformula

If $\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^2)\otimes \mathbb{R}^2$ then $\pform(\psi,\psi)$ makes sense for $\psi\in C_0^\infty(\mathbb{R}^2)\otimes \mathbb{C}^2$. We define the {\em minimal} Pauli form $\pform_{\text{min}}$ as
\startformula
\startalign
\NC \dom(\pform_{\text{min}}) \NC = C_0^\infty(\mathbb{R}^2)\otimes \mathbb{C}^2;\NR
\NC \pform_{\text{min}}(\psi,\psi) \NC = \pform(\psi,\psi),\quad \psi\in\dom(\pform_{\text{min}}).\NR
\stopalign
\stopformula
It is closable and thus a self-adjoint operator $\pauli_{\text{min}}$ can be defined. We also define the {\em maximal} Pauli form $\pform_{\text{max}}$ as
\placeformula
\startformula
\startalign
\NC \dom(\pform_{\text{max}}) \NC = \bigl\{\,\psi\in L_2(\mathbb{R}^2)\otimes \mathbb{C}^2\bigm| \pform(\psi,\psi)<\infty\,\bigr\};\NR[keq:pformmax]
\NC \pform_{\text{max}}(\psi,\psi) \NC = \pform(\psi,\psi),\quad\psi\in\dom(\pform_{\text{max}}).\NR
\stopalign
\stopformula
In the presence of AB solenoids, $\vec{a}$ does not belong to $L_{2,\text{loc}}(\mathbb{R}^2)\otimes \mathbb{R}^2$. It was proved in~\cite[so] that the Pauli form can not be defined on smooth compactly supported $\psi$ via~\eqref[keq:pform] in this case. The way out of this is to redefine the Pauli form $\pform$ by an expression that makes sense even in this more singular case. This is done in~\cite[ervo] by writing the operators $\ann$ and $\cre$ as
\placeformula[keq:anncre]
\startformula
\ann =-2i e^{W}\PD{}{\bar{z}} e^{-W}\quad\text{and}\quad \cre = -2i e^{-W}\PD{}{z} e^{W},
\stopformula
and noting that the quadratic form
\placeformula[keq:pformw]
\startformula
\pform(\psi,\psi) = 4\int_{\mathbb{R}^2} \Big|\PD{}{\bz}\left(e^{-W}\psip\right)\Big|^2e^{2W}+\Big|\PD{}{z}\left(e^{W}\psim\right)\Big|^2 e^{-2W}\ed m(x),
\stopformula
$\psi=(\psip,\psim)^t$ makes sense even with this more singular field. If $\pform$ is defined on a maximal domain in the same way as in~\eqref[keq:pformmax], it yields a self-adjoint operator even with this singular field, usually called the {\em maximal} Pauli operator. The forms in~\eqref[keq:pform] and~\eqref[keq:pformw] coincides for more regular fields.


\page

\placeformula[keq:pauli]
\startformula
\pauli = 
\startpmatrix
\NC H-\frac{g}{2}B \NC 0 \NR
\NC 0 \NC H+\frac{g}{2}B\NR
\stoppmatrix.
\stopformula

\placeformula
\startformula
-\Delta W = B.
\stopformula
We set $\Pi_j = -i\PD{}{x^j}-a_j$ and
\startformula
 \ann = \Pi_1 - i \Pi_2,\quad \cre = \Pi_1 + i\Pi_2,
\stopformula

\placeformula[keq:komm]
\startformula
\ann\cre=\cre\ann+2B,\quad H=\cre\ann+B=\ann\cre-B.
\stopformula

\placeformula[keq:paulig]
\startformula
\pauli = 
\startpmatrix
\NC \cre\ann-\frac{g-2}{2}B \NC 0 \NR
\NC 0 \NC \ann\cre+\frac{g-2}{2}B\NR
\stoppmatrix.
\stopformula

\placeformula[keq:pauliw]
\startformula
\pauli = 
\startpmatrix
\NC \cre\ann \NC 0 \NR
\NC 0 \NC \ann\cre\NR
\stoppmatrix.
\stopformula

\placeformula[keq:paulia]
\startformula
\pauli = \dirac^2 = \Bigl(\sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\big)\Bigr)^2 = 
\startpmatrix
\NC 0 \NC \cre \NR
\NC \ann \NC 0\NR
\stoppmatrix^2
\stopformula

\placeformula[keq:pform]
\startformula
\pform(\psi,\psi)= \int_{\mathbb{R}^2} \Big|\sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\big)\psi\Big|^2\ed m(x).
\stopformula

\startformula
\startalign
\NC \dom(\pform_{\text{min}}) \NC = C_0^\infty(\mathbb{R}^2)\otimes \mathbb{C}^2;\NR
\NC \pform_{\text{min}}(\psi,\psi) \NC = \pform(\psi,\psi),\quad \psi\in\dom(\pform_{\text{min}}).\NR
\stopalign
\stopformula

\placeformula
\startformula
\startalign
\NC \dom(\pform_{\text{max}}) \NC = \bigl\{\,\psi\in L_2(\mathbb{R}^2)\otimes \mathbb{C}^2\bigm| \pform(\psi,\psi)<\infty\,\bigr\};\NR[keq:pformmax]
\NC \pform_{\text{max}}(\psi,\psi) \NC = \pform(\psi,\psi),\quad\psi\in\dom(\pform_{\text{max}}).\NR
\stopalign
\stopformula

\placeformula[keq:anncre]
\startformula
\ann =-2i e^{W}\PD{}{\bar{z}} e^{-W}\quad\text{and}\quad \cre = -2i e^{-W}\PD{}{z} e^{W},
\stopformula

\placeformula[keq:pformw]
\startformula
\pform(\psi,\psi) = 4\int_{\mathbb{R}^2} \Big|\PD{}{\bz}\left(e^{-W}\psip\right)\Big|^2e^{2W}+\Big|\PD{}{z}\left(e^{W}\psim\right)\Big|^2 e^{-2W}\ed m(x),
\stopformula





\placeformula
\startformula
H = (-i\nabla-\vec{a})^2.
\stopformula
It is an unbounded operator, so one should be more specific about the domain $\dom(H)$. To be able to define $H$ on $C^\infty_0(\mathbb{R}^{n})$ it is sufficient that
\placeformula[keq:areq]
\startformula
\vec{a}\in L_{4,\text{loc}}(\mathbb{R}^{n})\otimes \mathbb{R}^{n},\quad \div\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^{n}).
\stopformula
This follows by expanding $H$ as
\startformula
H = -\Delta + i\div \vec{a}+2i\vec{a}\cdot\nabla+\vec{a}\cdot\vec{a}.
\stopformula

\placeformula[keq:sform]
\startformula
h(u,u)=\int_{\mathbb{R}^n} \left|(-i\nabla-\vec{a})u\right|^2\ed m(x).
\stopformula
Assuming that $\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^n)\otimes \mathbb{R}^n$, we can define two forms $h_{\min}$ and $h_{\max}$ as
\startformula
\startalign
\NC \dom(h_{\text{min}}) \NC = C_0^\infty(\mathbb{R}^n);\NR
\NC h_{\text{min}}(u,u)\NC =h(u,u),\quad u\in\dom(h_{\text{min}});\NR
\stopalign
\stopformula

\placeformula
\startformula
\startalign
\NC \dom(h_{\text{max}}) \NC = \bigl\{\,u\in L_2(\mathbb{R}^n)\bigm| h(u,u)<\infty\,\bigr\}\NR
\NC h_{\text{max}}(u,u)\NC =h(u,u),\quad u\in\dom(h_{\text{max}}).\NR
\stopalign
\stopformula

\placeformula
\startformula
e^{if}(-i\nabla-\vec{a}_1)^2 e^{-if} = (-i\nabla -\vec{a}_2)^2.
\stopformula

\placeformula
\startformula
B(x)=2\pi\alpha \delta(x) \ed x^1\wedge \ed x^2.
\stopformula

\placeformula
\startformula
\vec{a}(x)=\frac{\alpha}{|x|^2}\bigl(-x^2,x^1\bigr).
\stopformula

\placeformula
\startformula
\startalign
\NC h(u,u) \NC =\int_{\mathbb{R}^2} |(-i\nabla-\vec{a})u|^2\ed m(x) \geq \int_{|x|<1}|(-i\nabla-\vec{a})u|^2\ed m(x)\NR
\NC \NC = \int_{|x|<1}|\vec{a}|^2\ed m(x) = \int_{|x|<1}\frac{\alpha^2}{|x|^2}\ed m(x) = +\infty\NR
\stopalign
\stopformula

\placeformula
\startformula
u(re^{i\theta})\sim c_{-\alpha}r^{-\alpha}+c_{\alpha-1}r^{\alpha-1}e^{-i\theta}+c_{\alpha}r^{\alpha}+c_{1-\alpha}r^{1-\alpha}e^{-i\theta},\quad r\searrow 0, 
\stopformula

\placeformula
\startformula
N\bigl(\Lambda_q\pm\lambda,\mu_{\pm},H(\vec{a}_0,\pm V)\bigr) \sim \frac{|\log\lambda|}{\log|\log\lambda|},\quad\lambda \searrow 0,
\stopformula

Moreover, if $W_\infty=0$, then they are able to prove a similar formula for the higher Landau levels,
\startformula
\lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,q}^{\pm}-\Lambda_q\bigr)^{1/j} = \frac{B_0}{2}\kap(K)^2,\quad q=0,1,\ldots.
\stopformula

\placeformula
\startformula
\lambda_{1,q}^+\geq\lambda_{2,q}^+\geq 
\cdots,\qquad \lambda_{1,q}^-\leq\lambda_{2,q}^-\leq \cdots
\stopformula
be the eigenvalues of $H(\vec{a},\pm V)$ in $(\Lambda_{q},\Lambda_{q+1})$ (for~$+$) and $(\Lambda_{q-1},\Lambda_q)$ (for~$-$), and let $\kap(K)$ be the logarithmic capacity of the set $K$, see~\cite[lan]. Then, if the magnetic scalar potential can be written as $W=-\frac{B_0}4|z|^2+W_\infty$, where $W_\infty$ is bounded, and $V$ has support in a compact $K$, $V\geq c>0$ on $K$, it holds that
\startformula
\lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,0}^{\pm}-\Lambda_0)\bigr)^{1/j} = \frac{B_0}{2}\kap(K)^2.
\stopformula
Moreover, if $W_\infty=0$, then they are able to prove a similar formula for the higher Landau levels,
\startformula
\lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,q}^{\pm}-\Lambda_q\bigr)^{1/j} = \frac{B_0}{2}\kap(K)^2,\quad q=0,1,\ldots.
\stopformula

\placeformula
\startformula
\startalign
\NC N(-\infty,\Lambda_0,H)\NC \leq 2n,\NR
\NC N(\Lambda_q,\Lambda_{q+1},H)\NC \leq qn,\quad q=0,1,\ldots.\NR
\stopalign
\stopformula

\placeformula
\startformula
\lim_{j\to\infty} \bigl(j!(\lambda_{j,q}^{+}-\Lambda_q)\bigr)^{1/j} = \frac{B_0}{2}\kap(K)^2,\quad q=0,1,\ldots.
\stopformula

\placeformula
\startformula
\lim_{j\to\infty} \bigl(j!(\Lambda_q-\lambda_{j,q}^{-})\bigr)^{1/j} = \frac{B_0}{2}\kap(K)^2,\quad q=0,1,\ldots.
\stopformula

\placeformula[keq:gpauli]
\startformula
\pauli = 
\startpmatrix
\NC \vphantom{\frac{g-2}{2}B}\cre\ann \NC 0 \NR
\NC 0 \NC \vphantom{\frac{g-2}{2}B}\ann\cre\NR
\stoppmatrix
+
\startpmatrix
\NC -\frac{g-2}{2}B \NC 0 \NR
\NC 0 \NC \frac{g-2}{2}B\NR
\stoppmatrix,
\stopformula

\placeformula
\startproblem (This problem was proposed to me by Prof. Ari Laptev in private communication) Study Schrödinger operators with singular potentials in higher dimensions. For example, one could try to describe the self-adjoint extensions of the Schrödinger operator in $\mathbb{R}^{2d}$ with magnetic one-form
\startformula
a(x) = \sum_{j\neq k} \Phi_{j,k}\frac{x^{2k}-x^{2j}}{|z^j-z^k|^2}\ed x^{2j-1} + \Phi_{j,k} \frac{x^{2j-1}-x^{2k-1}}{|z^j-z^k|^2}\ed x^{2j}
\stopformula
initially defined on $C_0^\infty\bigl(\mathbb{R}^{2d}\setminus \{z^j=z^k\}_{j\neq k}\bigr)$.

The main difficulty is that the vector potential is singular, not just in one point as in two dimensions, but in all hyperplanes $z^j=z^k$. In two dimensions the extensions were described by certain singular boundary terms at the singular points. 
\stopproblem