\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 Schr��dinger 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 Schr��dinger 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 Schr��dinger operators with singular potentials in higher dimensions. For example, one could try to describe the self-adjoint extensions of the Schr��dinger 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



