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\begin{document}
\title{Persistent currents for 2D Schr{\"o}dinger operator
with a strong $\delta$-interaction on a loop}
\author{P.~Exner$^{a,b}$ and K.~Yoshitomi$^{c}$}
\date{}
\maketitle
\begin{quote}
{\small \em a) Department of Theoretical Physics, Nuclear Physics
Institute, \\ \phantom{e)x}Academy of Sciences, 25068 \v Re\v z,
Czech Republic \\
b) Doppler Institute, Czech Technical University,
B\v{r}ehov{\'a} 7,\\
\phantom{e)x}11519 Prague, Czech Republic \\
c) Graduate School of Mathematics, Kyushu University,
Hakozaki, \\ \phantom{e)x}Fukuoka 812-8581, Japan \\
\rm \phantom{e)x}exner@ujf.cas.cz,
yositomi@math.kyushu-u.ac.jp}
\vspace{8mm}
\noindent {\small We investigate the two-dimensional magnetic
Schr\"odinger operator $H_{B,\beta}=\left(-i\nabla-A\right)^2
-\beta\delta(\cdot-\Gamma)$, where $\Gamma$ is a smooth loop and
the vector potential $A$ corresponds to a homogeneous
magnetic field $B$
perpendicular to the plane. The asymptotics of negative
eigenvalues of $H_{B,\beta}$ for $\beta\to\infty$ is found. It shows, in
particular, that for large enough positive $\beta$ the system
exhibits persistent currents.}
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
One of the most often studied features of mesoscopic systems are
the persistent currents in rings threaded by a magnetic flux --
let us mention, e.g., \cite{CGR, CWB} and scores of other
theoretical and experimental papers where they were discussed.
For a charged particle (an electron) confined to a loop $\Gamma$
the effect is manifested by the dependence of the corresponding
eigenvalues $\lambda_n$ on the flux $\phi$ threading the loop,
conventionally measured in the units of flux quanta, $2\pi\hbar
c|e|^{-1}$. The derivative $\partial \lambda_n/\partial \phi$
equals $-{1\over c}I_n$, where $I_n$ is the persistent current
in the $n$--th state. In particular, if the particle motion on
the loop is free, we have
% ------------- %%
\begin{equation} \label{ideal}
\lambda_n(\phi) = {\hbar^2\over 2m^*} \left( 2\pi\over
L\right)^2 (n+\phi)^2,
\end{equation}
% ------------- %%
where $L$ is the loop circumference and $m^*$ is the effective
mass of the electron, so the currents depend linearly on the
applied field in this case.
Of course, the above example is idealized assuming that the
particle is strictly confined to the loop. In reality boundaries
of a quantum wire are potential jumps at interfaces of different
materials. As a consequence, electrons can be found outside the
loop, even if not too far when we consider energies at which the
exterior is a classically forbidden region.
A reasonable model respecting the essentially one-dimensional
nature of quantum wires is a 2D Schr\"odinger operator with an
attractive $\delta$-interaction on an appropriate curve
$\Gamma$, or more generally, a planar graph. Since the
interaction support has codimension one, the Hamiltonian can be
defined through its quadratic form and the corresponding
resolvent can be written explicitly as a generalization of the
Birman-Schwinger theory \cite{BT, BEKS}. This leads to some
interesting consequences such as the existence of bound states
due to bending of an infinite and asymptotically straight curve
\cite{EI}.
A natural question is how such a model is related to the ideal
one in which the electron is strictly confined to the curve
$\Gamma$. In \cite{EY} we have derived an asymptotic formula
showing that if the $\delta$ coupling is strong, the negative
eigenvalues approach those of the ideal model in which the
geometry of $\Gamma$ is taken into account by means of an
effective curvature-induced potential. The purpose of this paper
is to ask a similar question in the situation when the electron
is subject in addition to a homogeneous magnetic field $B$
perpendicular to the plane. We are going to derive an analogous
asymptotic formula where now the presence of the magnetic field
is taken into account via the boundary conditions specifying the
domain of the comparison operator.
An easy consequence of this result is that for a strong enough
$\delta$-interaction the negative eigenvalues of our Hamiltonian
are not constant as functions of $B$, i.e. that the system
exhibits persistent currents. Their further properties depend,
of course, on the specific shape of $\Gamma$; this fact as well
as the stability of such currents with respect to a disorder
raise questions about optimal ways of interpreting the
corresponding magnetic transport. We comment on this point in
the concluding remarks.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
\section{Description of the model and the results}
As we have said above we are going to study the Schr{\"o}dinger
operator in $L^2(\mathbb{R}^2)$ with a constant magnetic field
and an attractive $\delta$-interaction on a loop. For the sake
of simplicity we emply rational units, $\hbar =c= 2m^* =1$ and
absorb the electron charge into the field intensity $B$. We
shall use the circular gauge, $A(x,y)=\left(-{1\over
2}By,{1\over 2}Bx \right)$.
Let $\Gamma:[0,L]\owns s\mapsto (\Gamma_{1}(s),\Gamma_{2}(s))\in
\mathbb{R}^{2}$ be a closed counter-clockwise $C^{4}$ Jordan
curve which is parametrized by its arc length. Given $\beta>0$
and $B\in\mathbb{R}$, we define
% ------------- %%
$$ %\begin{equation} \label{}
q_{B,\beta}[f]=\left\| \left(-i\partial_x +{1\over 2}By \right)f
\right\|^{2}
+\left\|\left(-i\partial_y -{1\over 2}Bx\right)f\right\|^{2} -\beta
\int_{\Gamma}|f(x)|^{2}\,ds
$$ %\end{equation}
% ------------- %%
with the domain $H^{1}(\mathbb{R}^{2})$, where $\partial_x
\equiv \partial/\partial x$ etc., and the norm refers to
$L^2(\mathbb{R}^2)$. It is straightforward to check that the
form $q_{B,\beta}$ is closed and below bounded. We denote by
$H_{B,\beta}$ the self-adjoint operator associated with it which
can be formally written as
% ------------- %%
$$ %\begin{equation} \label{}
H_{B,\beta}=\left(-i\nabla-A\right)^2
-\beta\delta(\cdot-\Gamma)\,.
$$ %\end{equation}
% ------------- %%
Our main aim is to study the asymptotic behavior of the
negative eigenvalues of $H_{B,\beta}$ as $\beta\to +\infty$.
Let $\gamma:[0,L]\owns s\mapsto
(\Gamma_{1}^{\prime\prime}\Gamma_{2}^{\prime}
-\Gamma_{2}^{\prime\prime}\Gamma_{1}^{\prime})(s) \in\mathbb{R}$
be the signed curvature of $\Gamma$. We denote by $\Omega$ the
region enclosed by $\Gamma$, with the area $|\Omega|$, and
define the operator
% ------------- %%
$$ %\begin{equation} \label{}
S_{B}=-\frac{d^{2}}{ds^{2}}-\frac{1}{4}\gamma(s)^{2}
$$ %\end{equation}
% ------------- %%
on $L^{2}((0,L))$ with the domain
% ------------- %%
$$ %\begin{equation} \label{}
P_{B}=\{\,\varphi\in H^{2}((0,L));\quad
\varphi^{(k)}(L)=\exp(-iB|\Omega|)\varphi^{(k)}(0),\;
k=0,1\,\}\,,
$$ %\end{equation}
% ------------- %%
where $\varphi^{(k)}$ stands for the $k$-th derivative.
We fix $j\in\mathbb{N}$ and denote by $\mu_{j}(B)$ the $j$-th
eigenvalue of $S_{B}$ counted with multiplicity. Our main
results is the following claim.
% ------------- %
\begin{theorem} \label{main}
Let $n$ be an arbitrary integer
and let $\emptyset\neq I\subset\mathbb{R}$
be a compact interval. Then there exists
$\beta(n,I)>0$ such that
% ------------- %%
$$ %\begin{equation} \label{}
\sharp\{ \sigma_{d}(H_{B,\beta})\cap (-\infty,0)\}\geq n
\quad\mathrm{for}\quad \beta\geq \beta(n,I) \quad\mathrm{and}\quad
B\in I\,.
$$ %\end{equation}
% ------------- %%
For $\beta\geq \beta(n)$ and $B\in I$ we denote by
$\lambda_{n}(B,\beta)$ the $n$-th eigenvalue of $H_{B,\beta}$
counted with multiplicity. Then $\lambda_{n}(B,\beta)$ admits an
asymptotic expansion of the form
% ------------- %%
$$ %\begin{equation} \label{}
\lambda_{n}(B,\beta)= -\frac{1}{4}\beta^{2}+\mu_{n}(B)
+\mathcal{O}(\beta^{-1}\ln\beta)\quad\mathrm{as}\quad
\beta\to+\infty\,,
$$ %\end{equation}
% ------------- %%
where the error term is uniform with respect to $B\in I$.
\end{theorem}
% ------------- %
Recall that the flux $\phi$ through the loop is $B|\Omega|/2\pi$
in our units. The existence of persistent currents is then given
by the following consequence of the above result.
% ------------- %
\begin{corollary} \label{nonconst}
Let $\emptyset\neq I\subset\mathbb{R}$
be a compact interval and let $n\in\mathbb{N}$. Then there
exists a constant $\beta_{1}(n,I)>0$ such that the function
$\lambda_{n}(\cdot,\beta)$ is not constant for
$\beta\geq \beta_{1}(n,I)$.
\end{corollary}
% ------------- %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
\section{The proofs}
Since the spectral properties of $H_{B,\beta}$ are clearly
invariant with respect to Euclidean transformation of the plane, we may
assume without any loss of generality that the curve $\Gamma$
parametrizes in the following way,
% ------------- %%
$$ %\begin{equation} %\label{}
\Gamma_{1}(s) = \Gamma_{1}(0)+\int^{s}_{0}\cos
H(t)\,dt\,, \quad
\Gamma_{2}(s) = \Gamma_{2}(0)+\int^{s}_{0}\sin H(t)\,dt\,,
$$ %\end{equation}
% ------------- %%
where $H(t):=-\int^{t}_{0}\gamma(u)\,du $. Let $\Psi_{a}$ be the
map
% ------------- %%
$$ %\begin{equation} %\label{}
\Psi_{a}:\, [0,L)\times (-a,a)\owns (s,u)\mapsto
(\Gamma_{1}(s)\!-\!u\Gamma_{2}^{\prime}(s),
\Gamma_{2}(s)\!+\!u\Gamma^{\prime}_{1}(s))\in\mathbb{R}^{2}.
$$ %\end{equation}
% ------------- %%
By \cite[Lemma 2.1]{EY} we know that there exists $a_{1}>0$ such
that the map $\Psi_{a}$ is injective for all $a\in (0,a_{1}]$.
We fix thus $a\in (0,a_{1})$ and denote by $\Sigma_{a}$ the
strip of width $2a$ enclosing $\Gamma$
% ------------- %%
$$ %\begin{equation} %\label{}
\Sigma_{a}:=\Psi_{a}\left([0,L)\times (-a,a)\right)\,.
$$ %\end{equation}
% ------------- %%
Then the set $\mathbb{R}^{2}\backslash\Sigma_{a}$ consists of
two connected components which we denote by
$\Lambda^{\mathrm{in}}_{a}$ and $\Lambda^{\mathrm{out}}_{a}$,
where the interior one, $\Lambda^{\mathrm{in}}_{a}$, is compact.
We define a pair of quadratic forms,
% ------------- %%
\begin{eqnarray*} %\label{}
\lefteqn{ q_{B,a,\beta}^{\pm}[f]=\left\Vert \left(-i\partial_x
+{1\over 2}By \right)f \right\Vert^{2}_{L^{2}(\Sigma_{a})}
+\left\Vert \left(-i\partial_y-{1\over 2}Bx \right)f
\right\Vert^{2}_{L^{2}(\Sigma_{a})} } \\ &&
\phantom{AAAA} -\beta\int_{\Gamma}|f(x)|^{2}\,ds \,,
\phantom{AAAAAAAAAAAAAAAAAAAA}
\end{eqnarray*}
% ------------- %%
which are given by the same expression but differ by their
domains; the latter is $H_0^{1}(\Sigma_{a})$ for
$q_{B,a,\beta}^{+}$ and $H^{1}(\Sigma_{a})$ for
$q_{B,a,\beta}^{-}$. Furthermore, we introduce the quadratic
forms
% ------------- %%
$$ %\begin{equation} \label{}
e_{B,a}^{j,\pm}[f]=\left\Vert \left(-i\partial_x
+{1\over 2}By \right)f \right\Vert^{2}_{L^{2}(\Lambda^j_{a})}
+\left\Vert \left(-i\partial_y-{1\over 2}Bx \right)f
\right\Vert^{2}_{L^{2}(\Lambda^j_{a})}
$$ %\end{equation}
% ------------- %%
for $j=\mathrm{out, in}$, with the domains
$H^{1}_{0}(\Lambda^j_{a})$ and $H^{1}(\Lambda^j_{a})$
corresponding to the $\pm$ sign, respectively. Let
$L^{\pm}_{B,a,\beta}$, $E^{ \mathrm{out},\pm}_{B,a}$, and
$E^{\mathrm{in},\pm}_{B,a}$ be the self-adjoint operators
associated with the forms $q^{\pm}_{B,a,\beta}$, $e^{
\mathrm{out},\pm}_{B,a}$, and $e^{ \mathrm{in},\pm}_{B,a}$,
respectively.
As in \cite{EY} we are going to use the Dirichlet-Neumann
bracketing with additional boundary conditions at the boundary of
$\Sigma_a$. It works in the magnetic case too as one can see
easily comparing the form domains of the involved operators --
cf.~\cite[Thm.~XIII.2]{RS}. We get
% ------------- %%
\begin{equation} \label{DNest}
E^{ \mathrm{in},-}_{B,a}\oplus L^{-}_{B,a,\beta}\oplus E^{
\mathrm{out},-}_{B,a}\leq H_{B,\beta}\leq E^{ \mathrm{in},+}_{B,a}
\oplus L^{+}_{B,a,\beta}\oplus E^{ \mathrm{out},+}_{B,a}
\end{equation}
% ------------- %%
with the decomposed estimating operators in $L^2(\mathbb{R}^2)=
L^{2}(\Lambda^{ \mathrm{in}}_{a})\oplus L^{2}(\Sigma_{a})\oplus
L^{2}(\Lambda^{ \mathrm{out}}_{a})$. In order to assess the
negative eigenvalues of $H_{B,\beta}$, it suffices to consider
those of $L^{+}_{B,a,\beta}$ and $L^{-}_{B,a,\beta}$, because the
other operators involved in (\ref{DNest}) are positive. Since the
loop is smooth, we can pass inside $\Sigma_a$ to the natural
curvilinear coordinates: we put
% ------------- %%
$$ %\begin{equation} \label{}
(U_{a}f)(s,u)=(1+u\gamma(s))^{1/2}f(\Psi_{a}(s,u))
\quad\mathrm{for}\quad f\in L^{2}(\Sigma_{a})
$$ %\end{equation}
% ------------- %%
which defines the unitary operator $U_{a}$ from
$L^{2}(\Sigma_{a})$ to $L^{2}((0,L)\times (-a,a))$. To express the
estimating operators in the new variables, we introduce
% ------------- %%
\begin{eqnarray*} %\label{}
Q_{a}^{+} &\!=\!& \Big\{\, \varphi\in H^{1}((0,L)\times
(-a,a));\quad \varphi(L,\cdot)=\varphi(0,\cdot) \quad
\mathrm{on}\quad (-a,a),\\ && \phantom{aAAAAAAAAAAAAAA}
\varphi(\cdot ,a)=\varphi(\cdot, -a)=0 \quad\mathrm{on}\quad (0,L)
\Big \}, \\ Q_{a}^{-} &\!=\!& \Big\{\,\varphi\in H^{1}((0,L)\times
(-a,a)); \quad \varphi(L,\cdot)=
\varphi(0,\cdot)\quad\mathrm{on}\quad (-a,a) \Big\},
\end{eqnarray*}
% ------------- %%
and define the quadratic forms
% ------------- %%
\begin{eqnarray} \label{estform}
\lefteqn{ b^{\pm}_{B,a,\beta}[g]} \\ &&
=\int^{L}_{0}\int^{a}_{-a}(1+u\gamma(s))^{-2} \left|\partial_s
g\right|^{2}\,du\,ds +\int^{L}_{0}\int^{a}_{-a} \left|\partial_u
g\right|^{2}\,du\, ds \nonumber \\ &&
+\int^{L}_{0}\int^{a}_{-a}V(s,u)|g|^{2}\,ds\,du
-\beta\int^{L}_{0}|g(s,0)|^{2}\,ds \nonumber \\ &&
-\frac{b_{\pm}}{2}\int^{L}_{0}
\frac{\gamma(s)}{1+a\gamma(s)}|g(s,a)|^{2}\,ds
+\frac{b_{\pm}}{2}\int^{L}_{0}
\frac{\gamma(s)}{1-a\gamma(s)}|g(s,-a)|^{2}\,ds \nonumber \\ && +
{1\over 4} \int^{L}_{0}\int^{a}_{-a}
B^{2}(\Gamma_{1}^{2}-2u\Gamma_{1}\Gamma_{2}^{\prime}
+\Gamma_{2}^{2}+2u\Gamma_{2}\Gamma_{1}^{\prime}+u^{2})|g|^{2}\,du\,ds
\nonumber \\ && +B\,{\mathrm{Im}}\int^{L}_{0}\int^{a}_{-a}
(\Gamma_{2}+u\Gamma_{1}^{\prime}) \left((1+u\gamma)^{-1}\cos
H\,\overline{g}\partial_s g -\sin H\,\overline{g}\partial_u g
\right)\,du\,ds \nonumber \\ &&
-B\,{\mathrm{Im}}\int^{L}_{0}\int^{a}_{-a}
(\Gamma_{1}-u\Gamma_{2}^{\prime}) \left( (1+u\gamma)^{-1}\sin H\,
\overline{g}\partial_s g +\cos H\,\overline{g}\partial_u
g\right)\,du\,ds \nonumber
\end{eqnarray}
% ------------- %%
on $Q^{\pm}_{a}$, respectively, where $b_+=0$ and $b_-=1$, and
% ------------- %%
$$ %\begin{equation} \label{}
V(s,u)= \frac{1}{2}(1+u\gamma(s))^{-3}u\gamma^{\prime\prime}(s)
-\frac{5}{4}(1+u\gamma(s))^{-4}u^{2}\gamma^{\prime}(s)^{2}
-\frac{1}{4}(1+u\gamma(s))^{-2}\gamma(s)^{2}
$$ %\end{equation}
% ------------- %%
is the well-known curvature-induced effective potential \cite{ES}.
Let $D^{\pm}_{B,a,\beta}$ be the self-adjoint operators associated
with the forms $b^{\pm}_{B,a,\beta}$, respectively. In analogy
with \cite[Lemma 2.2]{EY}, we get the following result.
% ------------- %
\begin{lemma} \label{curvilin}
$\:U^{*}_{a}D^{\pm}_{B,a,\beta}U_{a}=L^{\pm}_{B,a,\beta}.$
\end{lemma}
% ------------- %
The presence of the magnetic field gave rise to terms containing
$\overline{g}\partial_s g$ and $\overline{g}\partial_u g$ in
(\ref{estform}). In order to eliminate the corresponding
coefficients modulo small errors, we employ another unitary
operator. We put
% ------------- %%
$$ %\begin{equation} \label{}
T_{B}(s)=-{1\over 2} B\int^{s}_{0}
\left(\Gamma_{2}(t)\Gamma_{1}^{\prime}(t)
-\Gamma^{\prime}_{2}(t)\Gamma_{1}(t)\right)\,dt\,;
$$ %\end{equation}
% ------------- %%
it follows from the Green theorem that $T_{B} (L)=B|\Omega|\,$.
Then we define
% ------------- %%
$$ %\begin{equation} \label{}
(M_{B}h)(s,u):=\exp\Big\lbrack iT_{B}(s)+{i\over 2}Bu
\left(\Gamma_{2}(s)\sin H(s)+\Gamma_{1}(s)\cos H(s)\right)
\Big\rbrack\,h(s,u)
$$ %\end{equation}
% ------------- %%
for any $h\in L^{2}((0,L)\times (-a,a))$; it is straightforward to
check that $M_{B}$ is a unitary operator on $L^{2}((0,L)\times
(-a,a))$. We define
% ------------- %%
\begin{eqnarray*} %\label{}
\tilde{Q}_{B,a}^{+} &\!=\!& \Big\{\, \varphi\in H^{1}((0,L)\times
(-a,a));\quad \varphi(L,\cdot)=
\mathrm{e}^{-iB|\Omega|}\varphi(0,\cdot) \; \mathrm{on}\;
(-a,a),\\ && \phantom{aAAAAAAAAAAAAAAAA} \varphi(\cdot
,a)=\varphi(\cdot, -a)=0 \quad\mathrm{on}\quad (0,L) \Big \}, \\
\tilde{Q}_{B,a}^{-} &\!=\!& \Big\{\,\varphi\in H^{1}((0,L)\times
(-a,a)); \quad \varphi(L,\cdot)=\mathrm{e}^{-iB|\Omega|}
\varphi(0,\cdot) \;\mathrm{on}\;(-a,a) \Big\},
\end{eqnarray*}
% ------------- %%
and another pair of quadratic forms
% ------------- %%
\begin{eqnarray*} %\label{}
\lefteqn{ \tilde{b}^{\pm}_{B,a,\beta}[g]} \\ &&
=\int^{L}_{0}\int^{a}_{-a} \Big\{ (1+u\gamma)^{-2}|\partial_s
g|^{2} +|\partial_u g|^{2} \\ && + \big\lbrack
B(\Gamma_{2}+u\Gamma_{1}^{\prime})(1+u\gamma)^{-1}\cos H
-B(\Gamma_{1}-u\Gamma_{2}^{\prime})(1+u\gamma)^{-1}\sin H \\ &&
-B(1+u\gamma)^{-2}(\Gamma_{2}\cos H-\Gamma_{1}\sin H) \\ && +
B(1+u\gamma)^{-2}(\Gamma_{2}\sin H+\Gamma_{1}\cos H)^{\prime}u
\big\rbrack\, {\mathrm{Im}}(\overline{g}\partial_s g)
+W_{B}(s,u)|g|^{2} \Big\}\,du\,ds \\ &&
-\beta\int^{L}_{0}|g(s,0)|^{2}\,ds
-\frac{b_{\pm}}{2}\int^{L}_{0}\frac{\gamma(s)}{1+a\gamma(s)}
|g(s,a)|^{2}\,ds \\ &&
+\frac{b_{\pm}}{2}\int^{L}_{0}\frac{\gamma(s)}
{1-a\gamma(s)}|g(s,-a)|^{2}\,ds
\end{eqnarray*}
% ------------- %%
for $g\in\tilde{Q}^{\pm}_{B,a}$, respectively, where
% ------------- %%
\begin{eqnarray*} %\label{}
\lefteqn{ W_{B}(s,u)} \\ && =V(s,u)+ {1\over 4}
(1+u\gamma)^{-2}B^{2}u^{2}((\Gamma_{2}\sin H+\Gamma_{1} \cos
H)^{\prime})^{2} \\ && + {1\over 4}
B^{2}(\Gamma_{1}^{2}-2u\Gamma_{1}\Gamma_{2}^{\prime}
+\Gamma_{2}^{2}+2u\Gamma_{2}\Gamma_{1}^{\prime}+u^{2}) \\ &&
+B(\Gamma_{2}\!+\!u\Gamma_{1}^{\prime})(1+u\gamma)^{-1}
T_{B}^{\prime}(s)\cos H
-B(\Gamma_{1}\!-\!u\Gamma_{2}^{\prime})(1+u\gamma)^{-1}
T_{B}^{\prime}(s)\sin H \\ && + {1\over 4}
(1+u\gamma)^{-2}B^{2}(\Gamma_{2}\cos H-\Gamma_{1}\sin H)^{2} +
{1\over 4} B^{2}(\Gamma_{2}\sin H+\Gamma_{1}\cos H)^{2} \\ &&
+[B(\Gamma_{2}+u\Gamma_{1}^{\prime})(1+u\gamma)^{-1}\cos H
-B(\Gamma_{1}-u\Gamma_{2}^{\prime})(1+u\gamma)^{-1}\sin H \\ &&
-B(1+u\gamma)^{-2}(\Gamma_{2}\cos H-\Gamma_{1}\sin H)]\, {1\over
2} B(\Gamma_{2}\sin H+\Gamma_{1}\cos H)^{\prime}u \\ &&
+[-B(\Gamma_{2}+u\Gamma_{1}^{\prime})\sin H
-B(\Gamma_{1}-u\Gamma_{2}^{\prime})\cos H]\, {1\over 2}B
(\Gamma_{2}\sin H+\Gamma_{1}\cos H)\,.
\end{eqnarray*}
% ------------- %%
Let $\tilde{D}^{\pm}_{B,a,\beta}$ be the self-adjoint operators
associated with the forms $\tilde{b}^{\pm}_{B,a,\beta}$,
respectively. By a straightforward computation, one can check the
following claim.
% ------------- %
\begin{lemma} \label{gaugeout}
$\:M_{B}^{*}D^{\pm}_{B,a,\beta} M_{B}=\tilde{D}^{\pm}_{B,a,\beta}\,.$
\end{lemma}
% ------------- %
The next step is to estimate $\tilde{D}^{\pm}_{B,a,\beta}$ by
operators with separated variables. Denoting
% ------------- %%
$$ %\begin{equation} \label{}
\gamma_{+:}=\max_{[0,L]}|\gamma(\cdot)|
$$ %\end{equation}
% ------------- %%
we put
% ------------- %%
\begin{eqnarray*} %\label{}
N_{B}(a) &\!:=\!& \max_{(s,u)\in [0,L]\times [-a,a]} \Big|
B(\Gamma_{2}+u\Gamma_{1}^{\prime})(1+u\gamma)^{-1}\cos H \\ &&
-B(\Gamma_{1}-u\Gamma_{2}^{\prime})(1+u\gamma)^{-1}\sin H \\ &&
-B(1+u\gamma)^{-2}(\Gamma_{2}\cos H-\Gamma_{1}\sin H) \\ &&
+B(1+u\gamma)^{-2}(\Gamma_{2}\sin H+\Gamma_{1}\cos H)^{\prime}u
\Big|
\end{eqnarray*}
% ------------- %%
and
% ------------- %%
$$ %\begin{equation} \label{}
M_{B}(a) :=\max_{(s,u)\in [0,L]\times [-a,a]} \left|
W_{B}(s,u)+\frac{1}{4}\gamma(s)^{2} \right|\,.
$$ %\end{equation}
% ------------- %%
Let $\emptyset\neq I\subset\mathbb{R}$ be a compact interval. Then
there is a positive $K$ such that
% ------------- %%
$$ %\begin{equation} \label{}
N_{B}(a)+M_{B}(a)\leq Ka \quad\mathrm{for}\quad
00$ such that
% ------------- %%
$$ %\begin{equation} \label{}
|\mu_{j}^{+}(B,a)-\mu_{j}(B)|+ |\mu_{j}^{-}(B,a)-\mu_{j}(B)|\leq
C(j)a
$$ %\end{equation}
% ------------- %%
holds for $B\in I$ and $00$ such that
% ------------- %%
$$ %\begin{equation} \label{}
\left\Vert U^{+}_{B,a} - \left\lbrack
(1-a\gamma_{+})^{-2}+\frac{1}{2}N_{B}(a) \right\rbrack S_{B}
\right\Vert \leq C_{1}a
$$ %\end{equation}
% ------------- %%
for $00$ such that
% ------------- %%
$$ %\begin{equation} \label{}
\left|\mu^{+}_{j}(B,a)-\mu_{j}(B)\right| \leq C_{2}a
$$ %\end{equation}
% ------------- %%
for $00$ such that
% ------------- %%
$$ %\begin{equation} \label{}
\left|\mu^{-}_{j}(B,a)-\mu_{j}(B)\right| \leq C_{3}a
$$ %\end{equation}
% ------------- %%
for $0\frac{8}{3}$. Then $T^{+}_{a,\beta}$ has
only one negative eigenvalue, which we denote by
$\zeta^{+}_{a,\beta}$. It satisfies the inequalities
% ------------- %%
$$ %\begin{equation} \label{}
-\frac{1}{4}\beta^{2}<\zeta^{+}_{a,\beta}<
-\frac{1}{4}\beta^{2}+2\beta^{2}\exp \left(-\frac{1}{2}\beta
a\right)\,.
$$ %\end{equation}
% ------------- %%
(b) Let $a\beta>8$ and $\beta>\frac{8}{3}\gamma_{+}$. Then
$T^{-}_{a,\beta}$ has a unique negative eigenvalue
$\zeta^{-}_{a,\beta}$, and moreover, we have
% ------------- %%
$$ %\begin{equation} \label{}
-\frac{1}{4}\beta^{2}-
\frac{2205}{16}\beta^{2}\exp\left(-\frac{1}{2}\beta a\right)
<\zeta^{-}_{a,\beta}<-\frac{1}{4}\beta^{2}\,.
$$ %\end{equation}
% ------------- %%
\end{proposition}
% ------------- %%
\vspace{1.2em}
\noindent Now we are ready to prove Theorem~\ref{main}. We put
$a(\beta) =6\beta^{-1}\ln\beta$. Let $\xi^{\pm}_{\beta,j}$ be
the $j$-th eigenvalue of $T^{\pm}_{a(\beta),\beta}$, by
Proposition~\ref{propEY} we have
% ------------- %%
$$ %\begin{equation} \label{}
\xi^{\pm}_{\beta,1}=\zeta^{\pm}_{a(\beta)\,,\beta},\quad
\xi_{\beta,2}^{\pm}\geq 0\,.
$$ %\end{equation}
% ------------- %%
>From the decompositions (\ref{decomp}) we infer that $\{
\xi^{\pm}_{\beta,j} +\mu_{k}^{\pm}
(B,a(\beta))\}_{j,k\in{\mathbb{N}}}$, properly ordered, is the
sequence of the eigenvalues of $\hat{H}^{\pm}_{ B,a(\beta),\beta}$
counted with multiplicity. Proposition~\ref{evest} gives
% ------------- %%
\begin{equation} \label{lowest}
\xi^{\pm}_{\beta,j}+\mu_{k}( B,a(\beta))\geq \mu_{1}^{\pm}(
B,a(\beta))=\mu_{1}(B)+\mathcal{O}(\beta^{-1}\ln\beta)
\end{equation}
% ------------- %%
for $B\in I$, $j\geq 2$, and $k\geq 1$, where the error term is
uniform with respect to $B\in I$. For a fixed $j\in\mathbb{N}$, we
put
% ------------- %%
$$ %\begin{equation} \label{}
\tau^{\pm}_{B,\beta,j}=\zeta^{\pm}_{a(\beta),\beta}
+\mu_{j}^{\pm}(B,a(\beta)).
$$ %\end{equation}
% ------------- %%
Combining Propositions~\ref{evest} and \ref{propEY} we get
% ------------- %%
\begin{equation} \label{tauasympt}
\tau^{\pm}_{B,\beta,j}=-\frac{1}{4}\beta^{2}
+\mu_{j}(B)+\mathcal{O}(\beta^{-1}\ln\beta)\quad\mathrm{as}
\quad\beta \to\infty\,,
\end{equation}
% ------------- %%
where the error term is uniform with respect to $B\in I$. Let us
fix now $n\in\mathbb{N}$. Combining (\ref{lowest}) with
(\ref{tauasympt}) we infer that there exists $\beta(n,I)>0$
such that the inequalities
% ------------- %%
$$ %\begin{equation} \label{}
\tau^{+}_{B,\beta,n}<0,\quad
\tau^{+}_{B,\beta,n}<\xi^{+}_{\beta,j}+\mu_{k}^{+}(
B,a(\beta)),\quad
\tau^{-}_{B,\beta,n}<\xi^{-}_{\beta,j}+\mu_{k}^{-}( B,a(\beta))
$$ %\end{equation}
% ------------- %%
hold for $B\in I$, $\beta\geq\beta(n,I)$, $j\geq 2$, and $k\geq
1$. Hence the $j$-th eigenvalue of $\hat{H}^{\pm}_{ B,a(\beta),
\beta}$ counted with multiplicity is $\tau^{\pm}_{B,\beta,j}$
for $B\in I$, $j\leq n$, and $\beta\geq \beta(n,I)$. Let $B\in
I$ and $\beta\geq \beta(n,I)$. We denote by $\kappa^{\pm}_{j}
(B,\beta)$ the $j$-th eigenvalue of $L^{\pm}_{B,a,\beta}$.
Combining our basic estimate (\ref{DNest}) with
Lemmas~\ref{curvilin} and \ref{gaugeout}, relations
(\ref{upper}) and (\ref{lower}), and the min-max principle, we
arrive at the inequalities
% ------------- %%
\begin{equation} \label{finalest}
\tau^{-}_{B,\beta,j}\leq\kappa^{-}_{j}(B,\beta)\quad
\mathrm{and}\quad \kappa^{+}_{j}(B,\beta)\leq\tau^{+}_{
B,\beta,j}\quad\mathrm{for}\quad 1\leq j\leq n\,,
\end{equation}
% ------------- %%
so we have $\kappa^{+}_{n}(B,\beta)<0<\inf
\sigma_{\mathrm{ess}}(H_{B,\beta})$. Hence the min-max principle
and (\ref{DNest}) imply that $H_{B,\beta}$ has at least $n$
eigenvalues in $(-\infty,\kappa^{+}_{n}(B,\beta)]$. Given $1\leq
j\leq n$, we denote by $\lambda_{j}(B,\beta)$ the $j$-th
eigenvalue of $H_{B,\beta}$. It satisfies
% ------------- %%
%\begin{equation}
$$\kappa_{j}^{-}(B,\beta)\leq\lambda_{j} (B,\beta)
\leq\kappa_{j}^{+} (B,\beta)\quad\mathrm{for}\quad 1\leq j\leq
n\,;$$
% \end{equation}
% ------------- %%
this together with (\ref{tauasympt}) and (\ref{finalest})
implies that
% ------------- %%
$$ %\begin{equation} \label{}
\lambda_{j}(B,\beta)=-\frac{1}{4}\beta^{2}
+\mu_{j}(B)+\mathcal{O}(\beta^{-1}\ln\beta)
\quad\mathrm{as}\quad\beta \to\infty\quad \mathrm{for} \quad
1\leq j\leq n\,,
$$ %\end{equation}
% ------------- %%
where the error term is uniform with respect to $B\in I$. This
completes the proof. \QED \vspace{1.2em}
\noindent{\sl Proof of Corollary~\ref{nonconst}}: By
\cite[Thm~XIII.89]{RS} no eigenvalue $\mu_{n}(\cdot)$ is
constant on $I$. This together with Theorem~\ref{main} yields
the claim. \QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}
\section{Concluding remarks}
The above corollary answers the question we posed in the
introduction as a mathematical problem showing that a ring with a
strong enough attractive $\delta$-interaction does exhibit
persistent currents. On the other hand, from the physical point of
view it would be bold to identify a mere non-constantness of the
eigenvalues with a genuine magnetic transport around the loop.
The problem is similar to other situation where an electron can be
transported in a magnetic field due to the presence of a
``guiding'' perturbation. A prime example are the edge currents
\cite{Ha, MS} which attracted a wave of mathematical interest
recently in connection with the problem of stability of the
transport with respect to perturbations. In case of a single edge
and a weak disorder a part of the absolutely continuous
spectrum survives \cite{BP, FGW, MMP} but the fact itself gives no
quantitative information about the transport. On the other hand, a
system with more than one edge may have no continuous spectrum at
all and still it has states in which electrons travel distances
much larger than the corresponding cyclotron radius \cite{FM}.
In our case it is clear, for instance, that the loop geometry
influences the transport substantially. If $\Gamma$ is a circle,
e.g., than up the $\mathcal{O}(\beta^{-1}\ln\beta)$ error the
persistent-current plot will have the ideal saw-tooth shape as we
can see from the relation (\ref{ideal}); one expects that the
eigenfunctions will be ``spread'' around the whole circle. In
contrast, if the loop is rather ``wiggly'' the one-dimensional
comparison operator $S_B$ contains an irregular effective
potential coming from the rapidly varying curvature, which may
cause -- depending on the strength of such a ``disorder'' -- that
the most part of the electron wavefunction will be concentrated in
(the vicinity of) a part of the loop only. The same may happen if
the loop curvature is slowly changing but a disorder potential is
added to the Hamiltonian.
To distinguish the situations with a {\em significant} transport,
one needs clear\-ly to understand better the sketched
``disordered'' cases which do not fall into this category. We
leave this problem to a future publication.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledments}
The research has been partially supported by GAAS and the Czech
Ministry of Education within the projects A1048101 and ME170. %
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\end{document}