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1991 Mathematics Subject Classification 35B05
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\begin{document}
%\null\vspace{4cm}
{\centering
\bfseries
SPECTRAL THEORY FOR THE DIHEDRAL GROUP.
\footnote{Supported by Ministerium f\"ur Bildung, Wissenschaft und
Kunst der Republik \"Osterreich}%
\footnote{Work supported by the European
Union TMR grant FMRX-CT 96-0001 }%
\footnote{Supported by the Austrian Science Foundation,
grant number P12864-MAT}
\footnote{Supported by NSF grant DMS-9971932}
\\[2\baselineskip]%
{\renewcommand\thefootnote{}%
\footnote{1991 Mathematics Subject Classification 35B05}}
\par
\mdseries
\scshape
\small
B. Helffer$^1$\\
M. Hoffmann-Ostenhof$^2$ \\
T. Hoffmann-Ostenhof$^{3,4}$ \\
N. Nadirashvili$^5$\\[2\baselineskip]
\par
\upshape
D\a'epartement de Math\a'ematiques, Universit\a'e Paris-Sud$^1$\\
Institut f\"ur Mathematik, Universit\"at Wien$^2$\\
Institut f\"ur Theoretische Chemie, Universit\"at Wien$^3$\\
International Erwin Schr\"odinger Institute for Mathematical Physics$^4$\\
Department of Mathematics, University of Chicago$^5$
\\[2\baselineskip]
\today
}
\begin{abstract}
Let $H=-\Delta+V$ be a two-dimensional Schr\"odinger operator defined on
a bounded domain $\Omega \subset \mathbb{R}^2$ with Dirichlet boundary
conditions on $\partial \Omega$. Suppose that $H$ commutes with the
actions of the dihedral group $\mathbb D_{2n}$, the group of the
regular $n$-gone.
We analyze completely the multiplicity of the groundstate eigenvalues
associated to the different symmetry subspaces related to the irreducible
representations of $\mathbb D_{2n}$. In particular we find that the
multiplicities of these groundstate eigenvalues equal the degree of
the corresponding irreducible representation. We also obtain an
ordering of these eigenvalues. In addition we analyze the
qualitative properties of the nodal sets of the corresponding ground
state eigenfunctions.
\end{abstract}
\section{Introduction and main results.} \label{Section1}
Let $\Omega \subset\mathbb {R}^2$ be a bounded domain with
$ C^{\infty}$-boundary $\partial\Omega$
and $V\in C^{\infty}(\overline\Omega)$ be real valued. We assume that
$\Omega$ is either homeomorphic to the disk or the annulus.
Let $a(x)$ be a real valued symmetric
positive definite 2 by 2 matrix with entries
$a_{ij}\in C^\infty(\overline\Omega)$. We consider
the elliptic operator
\begin{equation}\label{divergence}
H=-\sum_{i,j=1}^2\frac{\partial}{\partial x_i}a_{ij}(x)\frac{\partial}
{\partial x_j} +V(x)
\end{equation}
on $L^2(\Omega)$ with Dirichlet boundary conditions. Its quadratic form
is given with form domain $Q(H)=W_0^{1,2}(\Omega)$ by
\begin{equation}\label{form}
\langle\varphi\;,\; H\, \varphi \rangle =\int_\Omega \sum_{i,j=1}^2a_{ij}(x)
\frac{\partial \overline\varphi}{\partial x_i}
\frac{\partial \varphi}{\partial x_j}\;dx +\int_\Omega V(x)
|\varphi (x)|^2 \; dx\;.
\end{equation}
The operator domain of $H$ is $W^{2,2}(\Omega)\cap W_0^{1,2}(\Omega)$.\\
We consider the eigenvalue problem
\begin{equation}\label{eigenvalue}
Hu=\lambda u,\quad\text{ with $u=0$ in $\partial \Omega$}.
\end{equation}
There is a sequence of eigenvalues
$ \lambda^{1}<\lambda^{2}\le\dots\le \lambda^{k}\le\dots$
with corresponding eigenvectors $u^{k}$. By elliptic regularity
the $u^k\in C^\infty(\overline \Omega)$. Since $H$ is real, for any
complex valued eigenfunction eigenfunction $u$ also $\Re u$ and $\Im u$
are eigenfunctions. We shall denote the restriction of $Q(H)$
to real valued functions by $Q_\Re(H)$.
Of course, if $a_{ij}=\delta_{ij}$, then
\eqref{eigenvalue} becomes the standard eigenvalue problem for the
Dirichlet realization of the Schr\"odinger operator~:
\begin{equation}\label{Schroedinger}
(-\Delta+V)u=\lambda u\;.
\end{equation}
In the following we assume that $H$ and $\Omega$ are invariant with respect
to the actions of $\mathbb D_{2n}$, the group of the regular $n$-gone.
$\mathbb D_{2n}$ has $2n$ elements: $\{h_1,\dots, h_{2n}\}$,
$n$ rotations
$e,g,g^2,\dots, g^{n-1}$, where $e$ denotes the identity,
and $n$ reflections, $ T, Tg,\dots, Tg^{n-1}$.
As usual, (see for instance \cite{Serre:1967}), we
represent them by orthogonal $2\times 2$ matrices acting on $\mathbb R ^2$.
The element $g$ of the group is represented by a rotation by
$2\pi/n$ and the entries of the corresponding matrix are
\begin{equation}\label{rot}
g_{11}=g_{22}=\cos(2\pi/n), \quad g_{12}=-g_{21}=\sin(2\pi/n).
\end{equation}
The element $T$ is represented by the reflection $(x_1, x_2)\mapsto
(x_1,- x_2)$ with respect to the $x_1$-axis and
given by
\begin{equation}\label{refl}
T_{11}=- T_{22}= 1,\quad
T_{12}= T_{21}= 0.
\end{equation}
The elements $T$ and $g$ generate the group and
the multiplication law of $\mathbb D_{2n}$ can be worked out from
the relations
\begin{gather}\label{grouptable}
g^k T=T g^{n-k},\quad k=0,\dots,n\\
g^n=e\\
T^2=e.
\end{gather}
In polar coordinates $g$ acts on a point
$(x_1, x_2)= (r \cos \omega, r \sin \omega)$ in $\Omega$ by
$$ g(r,\omega)= (r,\omega+2\pi/n) \;,$$
and $ T$ by $$ T(r,\omega)=(r,-\omega)\;.$$
We define, for $ h\in \mathbb D_{2n}$ and $u\in L^2(\Omega)$, $hu$ by~:
\begin{equation*}
hu(x_1,x_2)=u(h^{-1}(x_1,x_2))\;,
\end{equation*}
where the $h$ are the orthogonal matrices defined above.
By an abuse of notation we do not distinguish between the group element
$h$ and the associated orthogonal matrix.
$\mathbb D_{2n}$ can be considered as a semidirect product of $\mathbb Z_n$ and
$\mathbb Z_2$. The rotations alone give $\mathbb Z_n$ and we
can also generate $\mathbb D_{2n}$ by the reflections.
The dimensions of the different irreducible representations
of $\mathbb D_{2n}$ are given, (see \cite{Serre:1967}), by~:\\
(i) Two one-dimensional irreducible representations and $(n-1)/2$
two -dimensional ones when {\bf n is odd}.\\
(ii) Four one-dimensional irreducible representations and $(n-2)/2$
two-dimensional ones when {\bf n is even}.\\
If $H$ and $\Omega$ are invariant with respect to the actions of
$\mathbb D_{2n}$, the quadratic form domain of $H$, $Q_\Re(H)$,
can be decomposed
into invariant subspaces $A_{(\cdot)}$ (defined below) associated to
the irreducible
representations of $\mathbb D_{2n}$ and we can write $H$ as a direct sum.
In the following the $A_{(\cdot)}$ will always denote invariant subspaces of
real valued functions.
This leads,
(see for example \cite{Serre:1967}), to the following symmetry subspaces.
\begin{equation}\label{symmetric}
A_{0,s}=\{u\in Q_\Re(H): g u =u \mbox{ and } T u = u \}\;.
\end{equation}
and
\begin{equation}\label{anti}
A_{0,a}=\{u\in Q_\Re(H): gu= u \mbox{ and } Tu = - u \}.
\end{equation}
For the case $n$ even we have two additional symmetry
subspaces.
But we consider first the symmetry subspaces associated to the
two-dimensional irreducible representations which are related to the
rotations.
%%and we make first a few remarks on $\mathbb Z_n$, the abelian
%% subgroup of $\mathbb D_{2n}$ which acts only by rotations.
%%
In the complex case these symmetry spaces can be parametrized
by $\ell$ with $ 0 < \ell < \frac n2$ and are defined by:
\begin{equation}
C_\ell = B_\ell \oplus B_{-\ell}\;,
\end{equation}
where
\begin{equation}\label {Bl}
B_\ell =\{w\in Q(H)\;:\; g w= e^{ 2\pi i\ell/n}w\}\;.
\end{equation}
$T$ maps $B_\ell$ onto $B_{-\ell}$
and so $C_\ell$ is a representation space of $\mathbb D_{2n}$.
Complex conjugation maps $B_\ell$ onto $B_{-\ell}$
and so $C_\ell$ can be recognized as the complexification
of the real space $A_\ell$
\begin{equation}\label{Al}
A_\ell=\{u\in Q_\Re(H): u-2\cos(2\ell\pi/n) g u+g^2u=0\}
\end{equation}
such that
\begin{equation}
C_\ell = A_\ell \otimes \mathbb C\;
\end{equation}
where \eqref{Al} follows from an easy computation based on \eqref{Bl}, see
also \cite{Serre:1967}.
Finally for $n$ even, $\ell=n/2$ plays a special role
since $e^{i\pi}=-1$ and this leads to the two other invariant
symmetry spaces defined by~:
\begin{equation}\label{An/21}
A_{n/2,s}=\{u\in Q_\Re(H): g u= - u,\quad
T u = u\},
\end{equation}
and
\begin{equation}\label{An/22}
A_{n/2,a}=\{u\in Q_\Re(H): g u = - u , \quad
T u=- u\}.
\end{equation}
The spaces $A_{(\cdot)}$
are invariant with respect to the actions of $D_{2n}$.
Let us now write $H$ into a direct sum of operators where
each $H_{(\cdot)}$ is the Friedrichs extension associated to the
quadratic form one obtains regarding $H$ with quadratic form domain
$ A_{(\cdot)}$.
\begin{proposition}\label{Hdirectsum}
Let $H$ and $\Omega$ be invariant under the actions of $\mathbb D_{2n}$.
Then
\begin{equation}\label{oddn}
H=H_{0,s}\oplus(\bigoplus_{\ell=1}^{(n-1)/2} H_{\ell})\oplus H_{0,a},
\text{ for {\bf n odd}}
\end{equation}
and
\begin{equation}\label{evenn}
H=H_{0,s}\oplus(\bigoplus_{\ell=1}^{n/2-1} H_{\ell})
\oplus H_{n/2,s}\oplus H_{n/2,a}
\oplus H_{0,a}\text{ for {\bf n even}}.
\end{equation}
\end{proposition}
For each of these operators, we have a spectrum and we
write for the corresponding eigenvalue problem
\begin{equation}\label{Hdot}
H_{(\cdot)}u^{i}_{(\cdot)}=\lambda_{(\cdot)}^i u_{(\cdot)}^i,
\quad i\in \mathbb N,\quad \lambda_{(\cdot)}^1
\le \lambda_{(\cdot)}^2\dots
\le \lambda_{(\cdot)}^k \le \dots.
\end{equation}
For simplicity we write for the groundstate eigenvalues
in the various symmetry subspaces $\lambda_{(\cdot)}=\lambda_{(\cdot)}^1$.
Also the eigenfunctions and eigenspaces associated to the
$\lambda_{(\cdot)}$ will be denoted by $u_{(\cdot)}$ and $U_{(\cdot)}$.
We can now formulate our main results.
\begin{theorem}.\label{Thmult}\\
The multiplicities of the eigenvalues $m(\lambda_{(\cdot)})$
satisfy
\begin{equation}\label{mSAs}
m(\lambda_{0,s})=1,
\end{equation}
\begin{equation}\label{mSAa}
m(\lambda_{0,a})=1
\end{equation}
and for $0<\ell4$.
\item
Note that for two-dimensional problems \eqref{eigenvalue} there are universal
bounds to $m(\lambda^{k})$ \cite{Cheng:1976,ColindeVerdiere:1986,
Besson:1988, Nadirashvili:1988, HoMiNa:1999, HoHoN:1999}. Thereby Courant's
nodal theorem plays a crucial role, see e.g. \cite{Chavel:1984}.
The situation here is quite different.
There could be, and this can be seen already for the radial case with
a potential, $k$ eigenvalues in the symmetry subspace $ A_{0,s}$ below
$\lambda_1$ for any given $k$.
\item We have chosen to treat in this paper the simplest non-trivial
case. In a forthcoming paper \cite{HeHoHoNa:2001}
we shall investigate related problems in a more general setting,
e.g. periodic Schr\"odinger operators and Aharonov-Bohm Hamiltonians.
\end{enumerate}
\end{remarks}
We finally indicate the structure of the proof of these two theorems.
In the next section we collect some observations concerning the representation
theory for $\mathbb D_{2n}$. In section 3 we describe some
direct spectral consequences of $\mathbb D_{2n}$-symmetry. In sections 4 and 5
we reduce the problem to a radially symmetric domain and then set up
a perturbation theory for which the radial case serves as the
unperturbed case. In section 6 we collect some general observations on nodal
sets of
eigenfunctions. In section 7 we analyze the nodal structure of eigenfunctions
of radial problems, introduce the notion of canonicity for the general
problem and state in Theorem 7.2 properties of nodal sets.
In section 8 we analyze the stability of the canonicity of nodal sets
with respect to perturbations.
In section 9 (this is the core of the proof) we prove the multiplicity
results by analyzing how this canonicity can be broken
and prove Theorem 7.2. Finally in section 10 we prove the ordering
of the eigenvalues using the previous analysis of the nodal sets and
some simple partial integration argument.
\section{Representation Theory.}\label{sRep}
\subsection{Basics.}
We first come back to the link between $C_\ell$, $B_\ell$ and $A_\ell $
appearing in the introduction.\\
The group $\mathbb D_{2n}$ contains a subgroup of order $n$, $\mathbb Z_n$
generated by $e$ and $g$ and it will be useful
to play between the representation theory of $\mathbb Z_{n} $
and $\mathbb D_{2n} $.\\
Let $0 < \ell < \frac n2 $. We recall that $A_\ell$ (resp. $C_\ell$)
is the real (resp. complex) invariant space corresponding
to an irreducible representation of $\mathbb D_{2n}$ of
degree~$2$ and $B _\ell $ corresponds to an irreducible
representation of $\mathbb Z_{n}$ of
degree $1$.\\
We have also mentioned the relation between these different spaces.
Let us be a little more explicit.
Given a normalized $u \in A_\ell$, there exists $v\in A_\ell$,
such that $w= u+ i v $
is in $B_\ell $. Moreover $v$ is unique since
there are no purely imaginary elements in $B_\ell$.
Let us mention further useful properties
of this decomposition.
Observing that $ B_\ell$
and $ B_{-\ell}$ are orthogonal, we get
\begin{equation}\label{orth}
\int_{\Omega_0}w(x)^2\,dx=0
\end{equation}
which is equivalent to
\begin{equation}\label{equiv}
\int_{\Omega_0}u(x)^2\,dx=\int_{\Omega_0}v(x)^2\, dx \text{ and }
\int_{\Omega_0}u(x) v(x)\, dx=0.
\end{equation}
In particular, $u$ and $v$ form an orthonormal basis for a
two-dimensional space, $\mathcal W_\ell $. Using (\ref{Bl}), we get indeed :
\begin{equation}\label{gaction}
g \, u =\cos(2\pi \ell /n) u -\sin (2\pi \ell /n) v
\end{equation}
and this shows that $gu$ belongs to $\mathcal W_\ell $ and is
not a multiple of $u$, due to the fact that $\sin( 2\pi \ell /n)\ne 0$.\\
As a corollary, we get~:
\begin{lemma}\label{Lemma3.2bis}
For $0<\ell}
\lambda_{0,a}>\max\{\lambda_{0,s},\max_{0<\ell\max\{\lambda_{n/2,s},\lambda_{n/2,a}\}.
\end{equation}
The other considerations will be based on the fact that the nodal sets
of $u_{0,a}$ and when $n$ is even of $u_{n/2,s}$, respectively, $u_{n/2,a}$
are enforced by symmetry.\\
Defining
\begin{equation}\label{tau}
\tau_j=\{x\in\overline \Omega:\omega=\pi j/n\},\quad j=0,\dots,2n-1,
\end{equation}
\eqref{anti} implies that
\begin{equation}\label{N0a}
\cup_{j=0}^{2n-1}\tau_j\subseteq\mathcal N(u_{0,a}).
\end{equation}
Similarily for $n$ even we obtain from \eqref{An/21} and \eqref{An/22}
\begin{equation}\label{Nn/2as}
\cup_{i=0}^{n-1}\tau_{2i}\subseteq \mathcal N(u_{n/2,a})\text{ and }
\cup_{i=1}^{n-1}\tau_{2i+1}\subseteq\mathcal N(u_{n/2,s}).
\end{equation}
Now in general every real valued eigenfunction of $H$ has the property
that its restriction to a nodal domain $D$ is the groundstate
of $H$ in this domain with Dirichlet boundary conditions. Furthermore
if we have a domain $D'\subset D$ with $\text{meas } (D\setminus D')>0$
then $\lambda'>\lambda$ where $\lambda$ and $\lambda'$ denote the
the groundstate eigenvalues of $H$ in $D$, respectively, $D'$
with Dirichlet boundary conditions.
We first show that $\lambda_{0,a},\lambda_{n/2,a}\lambda_{n/2,s}$ are
simple. Let $$\Omega^0=\{x\in\Omega:0<\omega<\pi/n\}$$ and suppose that
$\Phi$ is the associated groundstate of $H$ in $\Omega^0$ with
Dirichlet boundary conditions,
so that for some $\lambda^*\in\mathbb R$, $H\Phi=\lambda^*\Phi$
in $\Omega^0$. Extending $\Phi$ by 0 in
$\Omega\setminus \Omega^0$ we define
\begin{equation}\label{uaext}
\widetilde \Phi=\cup_{j=0}^{n-1}g^j(\Phi-T\Phi).
\end{equation}
It is easy to see that $\widetilde \Phi\in A_{0,a}$ and by construction
$\Omega^0$ is a nodal domain. The variational principle and the
uniqueness of groundstates hence implies that $\lambda^*=\lambda_{0,a}$
and that $m(\lambda_{0,a})=1$.
For $n$ even the arguments for $\lambda_{n/2,a},\lambda_{n/2,s}$ are
very similar. Indeed, let $\Omega^1=\{x\in \Omega:0<\omega<2\pi/n\}$,
and define $\Phi$ to be the groundstate of $H$ in $\Omega^1$ with Dirichlet
boundary conditions and extend it by 0 outside $\Omega^1$. Then
\begin{equation*}
\widetilde \Phi=\cup_{j=0}^{n-1}(-1)^jg^j\Phi\in A_{n/2,a}
\end{equation*}
and we can repeat the arguments from above to conclude that
$ m(\lambda_{n/2,a})=1$. The proof of the corresponding result for
$\lambda_{n/2,s}$ is almost identical.
To conclude this section we want to show \eqref{lambda0a>}.
Note first that $\Omega^0\subset \Omega^1$. Hence by the monotonicity
of groundstate eigenvalues with respect to domains
$\lambda_{0,a}>\lambda_{n/2,a}$. The proof that $\lambda_{0,a}>\lambda_{n/2,s}$
is almost identical. Indeed, for $u_{n/2,s}$, $\Omega^*=\{x\in \Omega:
\pi/n<\omega<3\pi/n\}$ is a nodal domain and contains strictly the nodal
domain $\{x\in\Omega:\pi/n<\omega<2\pi\}$ of $u_{0,a}$.
It remains to show
\begin{lemma}\label{l0a>ll}
$$\lambda_{0,a}>\lambda_\ell\text{ for } 0<\ell0$ sufficiently small.
\section{Perturbation theory for the interpolating family $H(\alpha)$. }
\label{sPert}
We are interested
in the multiplicity of the groundstate of $H(\alpha)$ restricted to $A_\ell$
for $\alpha=1$. We first introduce, for $0<\ellFrom now on, we use systematically Proposition \ref{PropIsosp} and
shall work with
$H_{\ell,a}$ instead of $H_\ell$.
It is then immediate that
\begin{equation}
I_\ell = \{\alpha\in [0,1+\epsilon_0): m(\lambda_{\ell,a}(\alpha))\ge 2\}\;.
\end{equation}
If $I_\ell\neq \emptyset$, let us define~:
\begin{equation}\label{alpha0}
\alpha_0(\ell)=\inf\{\alpha \in (0,1+\epsilon_0): m(\lambda_{\ell,a}(\alpha))
\ge 2\}\;,
\end{equation}
and, if $I_\ell=\emptyset$, we set:
\begin{equation}\label{alpha01}
\alpha_0(\ell)=1+\epsilon_0.
\end {equation}
The reference to ``antisymmetric'' will not be systematically repeated in this
section.
\begin{lemma}\label{Lemma3.1}
Let $0 < \ell < n/2$ and for $\alpha \in (-\epsilon_0,1+\epsilon_0)$,
let $P_\ell(\alpha)$ be the orthogonal projector on $U_\ell(\alpha)$.
Then the map $\alpha \mapsto P_\ell(\alpha)$ is $C^\infty$
for $\alpha \in [0,\alpha_0(\ell))$,
in the sense that, for any $u\in L^2$, $\alpha\mapsto P_\ell(\alpha)u$
is $C^\infty$ with value in $W^{k,2}(\Omega_0)$, for any $k\in \mathbb N$.\\
Furthermore we have a decomposition
\begin{equation*}
U_l(\alpha_0(\ell))=U_\ell^1(\alpha_0(\ell))\oplus U_\ell^2(\alpha_0(\ell))\,,
\end{equation*}
where
$$ \dim U_\ell^1(\alpha_0(\ell))= 1\,,$$
and
$$ U_\ell^1 (\alpha_0(\ell))= \lim_{\alpha \uparrow \alpha_0(\ell)}
U_\ell(\alpha)\;.
$$
The above limit is considered in the sense that, for any
$u\in L^2$, the vector $P_\ell(\alpha) u$
tends to $P_\ell^1 (\alpha_0(\ell))u$ as $\alpha \uparrow \alpha_0(\ell)$
in any
$W^{k,2} (\Omega_0)$ and hence in $C^\infty(\overline{\Omega_0})$. Here
$P_\ell^1(\alpha_0(\ell))$
is the orthogonal projector on $U_\ell^1(\alpha_0(\ell))$.
\end{lemma}
{\bf Proof of Lemma \ref{Lemma3.1}}.\\
We first note that $H(\alpha)$ can be extended as an analytic family of type A
in the sense of Kato \cite{Kato:1977}, see Chapter VII, Subsection 3.2.
Indeed we observe by standard elliptic theory that, for $\alpha$ in a complex
neighborhood of $[0,1]$, the operator domain $\mathcal D(H(\alpha))$ is
$W^{2,2}(\Omega_0)\cap W^{1,2}_0(\Omega_0)$ and that
$H^*(\alpha)=H(\overline\alpha)$. The statements are then
consequences of the standard Kato theory
(see \cite {Kato:1977, ReedSimon:1978}). The reduction to antisymmetric states
has permitted us to work with simple eigenvalues.
\finbox
As immediate consequence of Lemma~\ref{Lemma3.1} and using
what we explicitely know in the radial case (see section \ref{sCan} below), we get~:
\begin{lemma}\label{Lemma3.1bis}.\\
There exists an $\tilde{\alpha}_0 >0$ such that
$[0,\tilde{\alpha}_0] \cap I_\ell= \emptyset$ for $0<\ell0$ for boundary points.
\end{proposition}
Suppose $0\in \Omega_0$, hence $\Omega_0$ is the disk, then
for each $u\in A_\ell$,
$u\not\equiv 0$, we have in polar coordinates
\begin{equation}\label{radia}
u(r,\omega)= p_k(r,\omega)+ \mathcal{O}(r^{k+1})
\end{equation}
where
\begin{equation}\label{radiak}
k\in J_\ell:=\{|\ell+n\mathbb Z|\}.
\end{equation}
In particular,
\begin{equation}\label{radia1}
k \geq \ell\;, \text{ if } 0 < \ell 0
\end{equation}
and that, by the monotonicity of the coefficient $r^{-2}\ell^2$ with
respect to $\ell$,
\begin{equation}\label{ordering radial}
\lambda_\ell<\lambda_{\ell'},\text{ for } \ell < \ell'.
\end{equation}
Obviously we can write in the radial case
\begin{equation*}
H=\bigoplus_{\ell=0}^\infty H_\ell
\end{equation*}
as an infinite direct Hilbertian sum.\\
>From now on we will always denote the antisymmetric functions in
$ A_{\ell, a }$ by $u_\ell$ and the corresponding symmetric ones
in $ A_{\ell,s}$ by $v_\ell = \mathcal R_\ell \, u_\ell$, where
$\mathcal R_\ell $ is defined in (\ref{2.5})
(note that $\mathcal R_\ell \sin \ell \omega
= \cos \ell \omega $).
We consider for illustration the spherical case with respect to the
symmetries of an equilateral triangle and of the square, hence of
$\mathbb D_6$ and $\mathbb D_8$.
For $\mathbb D_6$ we have $u_0\in A_{0,s}$. There are two independent
functions
$u_1=f_1(r)\sin \omega$ and $v_1 = f_1(r)\cos\omega$ in $ A_1$.
Finally $u_3= f_3(r)\sin(3\omega)\in A_{0,a}$.
Note that also $u_2\in A_{1,a}$, but is not a
groundstate in this symmetry subspace.
Now consider $\mathbb D_8$. $u_1$ and $v_1$ belong as before to
$ A_{1,a}$, respectively $ A_{1,s}$.
$u_2=f_2(r)\sin(2\omega)$ belongs to $ A_{2,a}$ whereas
$v_2 =f_2(r)\cos(2\omega)$ belongs to $ A_{2,s}.$
$u_4=f_4(r)\sin(4\omega)$ is the groundstate in $ A_{0,a}$.
In our case here $u_2$ and $v_2$ yield the same eigenvalue,
but this will not be necessarily the case once we do not have
radial symmetry.
\subsection{Nodal patterns in the radial case.}
We proceed by analyzing the nodal patterns in greater detail.\\
We fix $0<\ell 0$ in $\Omega_0$.
In particular we have for $u_{\ell,rad} $, the antisymmetric function
\begin{equation}\label{radantiN}
\mathcal N(u_{\ell,rad})=\{x\in\overline{\Omega_0}:\omega=k\pi/\ell,
\quad k=0,\dots, 2\ell -1\}
\end{equation}
whereas for $v_{\ell,rad} = f_\ell (r)\cos \ell \omega$,
the symmetric function,
\begin{equation}\label{radsymN}
\mathcal N(v_{\ell,rad} )=\{x\in\overline{\Omega_0}:\omega=(2k+1)\pi/2\ell
\quad k=0,\dots, 2\ell -1.\}
\end{equation}
Of course, we recover in this particular case the general
results of Proposition \ref{Proposition2.4}.
\subsection{Nodal pattern for groundstates in $A_{\ell,a}$.}
We assume $0< \ell < \frac n2$ and
consider now a groundstate eigenfunction
$u_\ell\in A_{\ell,a}$ where we assume
that we have transformed the problem to $\Omega_0$. We recall
that we can associate to $u_\ell$
a symmetric eigenstate by~: $v_\ell = \mathcal R_\ell u_\ell$.
We also recall that by Proposition \ref{Proposition2.4}
\begin{equation}\label{Nuv}
\begin{array}{l}
{\bf C0 (\ell)} \quad \quad \quad \mathcal N_{\ell, a}
\subseteq \mathcal N(u_\ell).
\end{array}
\end{equation}
In addition to this property, we have observed above
that in the radial case the following
properties were satisfied:
{\bf C1}$(\ell)$\\
$\mathcal N(u_\ell)$,
consists of $2\ell$ non-intersecting nodal arcs
$\sigma_j$, $j=0,\dots 2\ell -1$, where
each of the $\sigma_j$ connects the outer and the inner boundary
in the case of the annulus. In the case of the disk we just
shrink the inner boundary to the origin $\{0\}$.
{\bf C2}$(\ell)$
\begin{equation}\label{can2}
\begin{split}
\mathcal N(u_\ell)\cap\mathcal N(\mathcal R_\ell u_\ell )=\emptyset
\text{ for the annulus}.\\
\text{For the disk } \mathcal N(u_\ell)\cap\mathcal N(\mathcal R_lu_\ell)
=\{0\}.\\
\end{split}
\end{equation}
{\bf C3}$(\ell)$\\
For the annulus and the disk the components of
$\mathcal N(u_\ell)\setminus \mathcal N_{\ell,a}$
are in the same $\Omega_{0j}$ as the corresponding
radial antisymmetric solution $u_{\ell,rad}$.
Our aim is to show that all these properties (together with {\bf C0$(\ell)$}
which is automatically satisfied) remain true for a general
antisymmetric ground state in $A_\ell$.
\begin{definition}{\bf (Canonicity).}\label{DefCan}\\
We say that $u_\ell\in A_{\ell,a}$ is {\bf canonical} if the properties
{\bf C1}$(\ell)$, {\bf C2}$(\ell)$ and {\bf C3}$(\ell)$ are satisfied.
\end{definition}
We are now able to state our result for the nodal sets of
eigenfunctions $u_\ell$ associated to the symmetry subspaces of
an operator $H$ as described in \eqref{divergence}.
\begin{theorem}\label{D2n}
Fix $n\ge 1$, then
\begin{equation}\label{NuS,As}
\mathcal N(u_{0,s})=\emptyset
\end{equation}
and
\begin{equation}\label{NuS,Aa}
\mathcal N(u_{0,a})=\cup_{j=0}^{2n-1}\tau_j.
\end{equation}
For $0<\ell0$ such that
$u_\ell(\alpha)$ is {\bf canonical} for $\alpha
\in [0,{\tilde \alpha}_1(\ell))$.
\end{lemma}
{\bf Proof of Lemma \ref{Lemma5.4}}:\\
Let $\beta_0 \in (-\epsilon_0,\alpha_0(\ell))$ such that $u_\ell(\beta_0)$
is canonical and denote by $U_{\ell,a} (\beta_0)$ (or more simply $U_\ell$)
the vector space generated by $u_\ell (\beta_0)$.\\
For $u\in U_\ell(\beta_0)\setminus \{0\}$, let us define $u(\cdot;\alpha)$ by~:
$$ u(\cdot;\alpha) = P_\ell(\alpha) u\;.$$
($P_\ell(\alpha)$ was defined in Lemma 5.1).
In particular $u(\cdot; \beta_0) = u$.\\
We want to show
the existence of $\epsilon >0$, such that $u(\cdot;\alpha)$ keeps
the same nodal pattern for $|\alpha - \beta_0| <\epsilon$.\\
{\bf The case of the annulus.}\\
We first observe, using Lemma 5.1, that
$(x,\alpha) \mapsto u(x;\alpha)$ is $C^\infty$
on $\overline{\Omega_0}\times (\beta_0-\epsilon,\beta_0 + \epsilon)$.
So by continuity
$\mathcal N (u(\cdot;\alpha))\cap\partial \Omega_0$ stays
for $|\alpha - \beta_0|<\epsilon$ ($\epsilon >0$ sufficiently small)
in a neighborhood of $\mathcal N (u)\cap \partial \Omega_0$.\\
Now, around each point $(x_0,\beta_0)$ with
$x_0\in \mathcal N(u)\cap \Omega_0$,
we can apply the standard implicit function theorem
to the function $(x,\alpha) \mapsto u(x;\alpha)$. We observe
indeed that $u(x_0;\beta_0) = 0$ and that $\nabla u(x_0;\beta_0) \neq 0$
(here we use Proposition~\ref{Proposition1.1}, assertions (ii) and (iii)).
Then in a small neighborhood of $x_0$ the zero set
of $ u(\cdot;\alpha)$ consists of a nodal line
(when $|\alpha -\beta_0|$ is small enough).
Now, for $x_0\in \partial \Omega_0 \setminus
(\mathcal N (u) \cap \partial \Omega_0$), we observe that
$(\partial u/\partial n) (x_0, \beta_0)\neq 0$ and this property
is stable under small perturbations. So no nodal line of
$u(\cdot;\alpha)$ can hit
the boundary outside a given neighborhood of
$\mathcal N (u) \cap \partial \Omega_0$, if $|\alpha -\beta_0|$
is small enough.
It is a bit more difficult to control the nodal set of $u(\cdot;\alpha)$
near $\mathcal N (u) \cap \partial \Omega_0 $.
Let us consider a point $x_0\in (\partial\Omega_0\cap \mathcal N(u;\beta_0))$
where one nodal arc hits the boundary.
We use local coordinates $(y_1,y_2)$ such that $x_0= (0,0)$ and
$\Omega_0$ is locally defined by $\{y\;|\;y_2 >0\}$. Let
$u(x;\alpha)= P_\ell (\alpha) u(x,\beta_0)$. In
the new coordinates $y$, we write~:
$\tilde{u} (y;\alpha) = u(x;\alpha)$.
This function is $C^\infty$ in $\alpha$ and $y$ (up to the boundary),
and we observe that ${\tilde u}(y_1,0; \alpha)= 0$ in a neighborhood
of $b_0$ and $\{y=(0,0)\}$.
Using Taylor's formula, there exists locally a $C^\infty$ function $g$ on
$\overline{\Omega_0}\times (\beta_0 -\epsilon, \beta_0 + \epsilon)$
such that~:
$$ {\tilde u}(y_1,y_2;\alpha) = y_2\, g(y_1,y_2;\alpha)\;.$$
We note that in $\Omega_0$ near $(0,0)$,
$\mathcal N(u(\cdot;\alpha))=\mathcal N ({\tilde u}(\cdot;\alpha))
= \mathcal N(g(\cdot;\alpha))$.\\
But let us compute $\nabla_y g(y_1,y_2;\alpha)$ at $(0,0;\beta_0)$.
The local description
of $u(\cdot;\beta_0)$ near $(0,0)$ shows (because $u(x)$
vanishes of first order at $x_0$) that
$|\nabla_y g (0,0;\beta_0)| \neq 0$. More precisely, we can see
that $(\partial_{y_1} g) (0,0;\beta_0)\neq 0$.\\
We can now apply the implicit function theorem. In order to apply it
in the standard form, we first
extend $g$ outside $\Omega_0$ and in a neighborhood of
$(0,0;\beta_0)$ as a $C^\infty$ function
of all the arguments. The implicit function theorem
implies then that the zeros of $g$
near $(0,0;\beta_0)$ are described by~:
$$\mathcal N(g(\cdot;\alpha)) =\{ (y_1,y_2)\;|\; y_1 =
\phi(y_2;\alpha) \text{ with } y_2 >0\}\;.$$
Here $\phi$ is a $C^\infty$ function on
$[0,\epsilon)\times (\beta_0 - \epsilon,\beta_0+\epsilon)$
with $\epsilon >0$ sufficiently small. This shows the local continuity of the nodal pattern
near the boundary for $\epsilon >0$ small enough.\\
{\bf The case of the disk}.\\
We proceed as in the case of the annulus away from
the origin. In a neighborhood of the origin, we can write, using
\eqref{radia} and \eqref{radia1}~:
$u(x_1,x_2;\alpha) = r^\ell v(r,\theta;\alpha)$
where $v$ is $C^\infty$ on $[0,r_0)\times S^1
\times (\beta_0 - \epsilon_0, \beta_0 + \epsilon_0)$ and $(r,\theta)$
correspond to the polar coordinates.\\
More precisely we know from \eqref{radia1}, that $u$ vanishes
at least to the order $\ell $ at the origin. Using Taylor's Formula
(with integral remainder)
for the function $[0,1]\ni t \mapsto f(t) = u(tx_1,tx_2)$, we get for
$f(1) = u(x_1,x_2)$ the expansion~:
$$ u(x_1,x_2) = \sum_{i+j=l} c_{i,j}(x_1,x_2;\alpha) x_1^i x_2^j\;.$$
We can then take~:
$$ v(r,\theta;\alpha)=
\sum_{i+j=l} c_{i,j}(r \cos \theta, r \sin \theta;\alpha) (\cos\theta)^i
(\sin \theta)^j\;.$$
Using the local description of $u(\cdot;\beta_0)$ and its canonicity,
we observe
that $\partial_\theta v (0, \theta_0;\beta_0) \neq 0$
where $\theta_0$ is a given angle
such that $v(0,\theta_0;\beta_0) =0$. The implicit function
theorem gives locally the existence
of a function $(r,\alpha) \mapsto \theta(r,\alpha)$ such that
$\theta(0,\beta_0) =\theta_0$
and $v(r, \theta(r,\alpha);\alpha)=0$. Moreover this map describes
all the zeros
in the neighborhood of the origin belonging to a sector~:
$\{r\exp i \theta_0\;|\; r >0\}$.
As mentioned above, we were dealing with a fixed canonical
$u\in U_\ell (\beta_0)\setminus \{0\}$.
We have shown that for $|\alpha-\beta_0|<\epsilon$ with $\epsilon$
sufficiently small
the nodal pattern of $u(x;\alpha)$ does not change.
Thus, since by assumption {\bf C1}$(\ell)$, {\bf C3}$(\ell)$ hold for
$u(x;\beta_0)$ these properties
hold also in a neighborhood of $\beta_0$. The proof of the
stability of property {\bf C2}$(\ell)$ also follows readily from these
observations.
\finbox
\section{Proof of Theorem 1.2 and of Theorem 7.2.}\label{sCrit}
In section 3 we have already shown that that $\lambda_{0,s},\lambda_{0,a}$ are
simple as well as $\lambda_{n/2,s},\lambda_{n/2,s}$. Thus it remains
to prove \eqref{mll} of Theorem 1.2 and \eqref{Ncano} of Theorem 7.2.
\subsection{On the critical values.}
We look at $H_{\ell,a}(\alpha)$ and the corresponding
lowest eigenvalue $\lambda_{\ell,a}(\alpha)$.
>From the analysis of the radial case and from Lemma \ref{Lemma5.3},
we know that $m(\lambda_{\ell,a}(\alpha))= 1$ for
$-\epsilon<\alpha <\epsilon$ for sufficiently
small $\epsilon$. We recall that $\alpha_0(\ell)$
was defined in (\ref{alpha0})-(\ref{alpha01})
and we now introduce another critical value corresponding to the
breaking of canonicity. We note that for $\alpha\in [0,\alpha_0(\ell))$,
the antisymmetric eigenfunction $u_\ell (\alpha)$ can be uniquely defined,
keeping the $C^\infty$ dependence with respect to $\alpha$,
if one adds the properties~:
$$
|| u_\ell (\cdot; \alpha) || = 1\; \text{ and } u_\ell (\cdot, \alpha) > 0
\mbox{ in }\Omega_{00}\;.$$
where $\Omega_{00}$ has been defined in \eqref{Omega0j}. We consequently
take this choice.
\begin{definition}
If there is some $\alpha >0$, for which $u_\ell(\alpha)$ is not canonical,
we define
\begin{equation}\label{alpha1l}
\alpha_{1}(\ell)=\inf\{0<\alpha<\alpha_0(\ell)\,:\;u_\ell
\text{ is not {\bf canonical}}\}.
\end{equation}
Otherwise we simply define it by~:
$$ \alpha_1(\ell) = \alpha_0(\ell)\;.$$
\end{definition}
%% {\bf Remark.}\\
%% One can also define analogously $\alpha_0(\ell,s)$ and $\alpha_1(\ell,s)$ or
%% more generally $\alpha_0(\ell)$ to be the smallest $\alpha$ such that
%% $m(\lambda_\ell)>2$ and $\alpha_1(\ell)$ to be the smallest
%% $\alpha\le\alpha_0(\ell)$ such that the corresponding eigenspace of $\lambda%% _\ell$
%% is no more canonical. We note also that by the definition of the canonicity
%% $\alpha_1(\ell)=\alpha_1(\ell,s)=\alpha_1(\ell,a)$ and correspondingly the d%% ifferent
%% $\alpha_0$ coincide for fixed $\ell$.
\begin{proposition}\label{alpha1=0}
Fix $0<\ellFrom the definition of $\alpha_1(\ell)$, we know that
$u_\ell (\alpha)$ is canonical 'in the past' (i.e. for
$\alpha < \alpha_1(\ell)$ ). This will imply some localization of
the nodal lines for $\alpha = \alpha_1(\ell)$.\\
$u_\ell (\alpha_1(\ell))$ still
satisfies, by 'former' canonicity (property {\bf C3}$(\ell)$),
the weaker property\\
{\bf C3'}$(\ell)$:\\
The components of $\mathcal N\left(u_\ell(\alpha_1(\ell))\right)
\setminus \mathcal N_{\ell,a}$
are in the same $\overline{\Omega_{0j}}$ as the corresponding
radial antisymmetric solution $u_{\ell,rad}$. $\mathcal N_{\ell,a}$
has been defined in (2.11).
\noindent Due to property {\bf C3}$(\ell)$ we have~:
\begin{equation}\label{Cons}
\text{ Two distinct nodal
lines cannot belong to two neighboring } \Omega_{0j}\;.
\end{equation}
This is an immediate consequence of the assumption that
$\ell < \frac n2$. But this remains true under
the weaker property {\bf C3'}$(\ell)$, so we get
{\bf C3''}$(\ell)$:\\
Two distinct nodal
lines of $u_\ell(\alpha_1(\ell) )$
cannot belong to two neighboring $\overline{\Omega_{0j}}$.
\noindent A similar argument shows that $\sigma_1$, (defined in section 7)
cannot belong to
$ \overline{\Omega_{00}} \cap \overline{\Omega_{01}}$. This implies~:
\begin{equation}\label{emptint}
\mathcal N ( u_{\ell}) \cap
\{ x\in \overline{\Omega_0}\;|\; \omega\in (0,2 \pi /n)\}
= \emptyset\;.
\end{equation}
Once these preliminary consequences of the 'former' canonicity are observed,
there are three
possible breakings of canonicity.
{\bf Case 1. Possible breaking of C1}($\ell$).\\
It is enough to consider nodal lines contained in some
$\overline{\Omega_{0j}}$.
In order to conclude that {\bf C1}$(\ell)$ is still satisfied for
$u_\ell (\alpha_1(\ell))$, it remains to show that two ``accidents''
cannot occur.
{\bf Case 1a}~: Self-intersection of a nodal line.\\
In this case there exists at least a nodal domain included
in $\Omega_{0,j}$ contradicting, as we have shown in section \ref{Section3},
that $\lambda_{0,a}>\lambda_\ell$.
{ \bf Case 1b}~: double (or higher) touching of a nodal line at the same
component of the boundary.\\
Again, a nodal domain has to be contained in an $\overline
{\Omega_{0,j}}$, and we get a similar contradiction.
In the case of the disk a 'double touching' of the origin means that
the order of vanishing of $u$ at $0$ must be $\ge \ell+1$, see \eqref{radiak}.
But as above two nodal lines cannot emanate from the origin and
stay within
one $\Omega_{0j}$ since this would again imply that there is a nodal domain
inside $\Omega_{0j}$.
{\bf Case 2. Possible breaking of C2}(l). \\
We now analyze the case when $\mathcal N \left(u_\ell(\alpha_1(\ell))\right)
\cap \mathcal N \left(\mathcal R_\ell u_\ell(\alpha_1(\ell))\right) $
is not empty. We recall that we have introduced $w_\ell = u_\ell + i v_\ell$
with $v_\ell = \mathcal R_\ell u_\ell$.\\
We now consider $x_0\neq 0$ such that (for short we write
$u= u_{\ell} (\alpha_1(\ell))\,$)
$$
u(x_0) = (gu)(x_0) = 0\;.
$$
One can distinguish two subcases.
{\bf Case 2a} $x_0$ lies on a line of reflection.\\
In this case, we have $n$ points obtained by successive rotations
of $2 \pi /n$ which are contained in $\mathcal N (w_\ell)$.
{\bf Case 2aa} \\
If $x_0=(r_0,\omega_0)$ and $\omega_0$ is an odd multiple of
$\frac \pi n$, we get
a contradiction to (\ref{emptint}): there is some point $g^k x_0$
such that the corresponding $g^k\omega_0$ is $\frac \pi n$.
{\bf Case 2ab} \\
If $x_0$ and the corresponding $\omega_0$ is an even multiple of $\frac \pi n$,
(\ref{emptint}) does not lead to an immediate contradiction.
The orbit of $x_0$,assume $x_0\neq 0$ for the disk,
is a subset of $\mathcal N (w_\ell)$
and we can assume that $x_0 \in \sigma_0$. By the assumption of 'former'
canonicity there is a strictly increasing sequence $j_t$
($t=0, \cdots , 2 \ell -1 $) such that
$j_0 =0$, $j_t < 2 n$ and such that $\sigma_k$
is contained in $\overline {\Omega_{0j_k}}$.
Moreover, we have (due to (\ref{Cons}) $j_{k+1}\geq j_k +2$.
This means that the $\overline{\Omega_{0j_k}}\setminus\{0\}$
are disjoint.
We have to show a contradiction to the property that $\mathfrak O (x_0)$
is in this set. But each $g^m x_0$, $m=0,..,n-1$, can only belong to one
of the sets $\overline{\Omega_{0,j_k}}$ and there are only
$2 \ell$ sets for $n$ different points. This gives the contradiction.
{\bf Case 2b} $x_0$ does not lie on a line of reflection.\\
Then we get easily a contradiction to (\ref{emptint}).
{\bf Case 3. Possible breaking of C3}($\ell$).\\
We consider the property {\bf C3''}($\ell$l). In order to violate
this property first a `` normal '' nodal line $\sigma_k $
must touch a $\tau_j$. Here we mean by normal that $\sigma_k$
is not a special antisymmetric line, that
is that $k n/\ell \not \in \mathbb Z$.
Let $x_0$ be the point where
some normal nodal line touches some reflection line $\tau_j$.
Then
\begin{equation}
T g^j x_0 =x_0\;.
\end{equation}
But $u$ is antisymmetric, so $T x_0 = g^j x_0$ belongs to $\mathcal N (u)$.
It remains to show that this implies that $\mathfrak O(x_0)
\subset\mathcal N (u)$. \\
{\bf But once we have shown $\mathbf{ \mathfrak O(x_0)\subset \mathcal N(u)}$
we are back to case 2.}\\
We have not used so far that $x_0$ is not on a special nodal line
of $u$.
This means that
\begin{equation}\label{condi}
(n -j) \ell /n \not \in \mathbb Z \text{ or }
j \ell/n \not \in \mathbb Z\;.
\end{equation}
In particular $j\neq 0$.
But we have by (\ref{gk})~:
$$
g^{-j} u = \sin (2 \pi \ell j /n)\, v + \cos(2 \pi\ell j /n)\, u \;.
$$
If $j\ell/n\not\in \mathbb Z$ we have to consider two cases.
For $$2\ell j/n\not\in \mathbb Z$$ $v(x_0)=0$ and $x_0\in \mathcal N(w)$.\\
For $$\ell j/n\in \frac{1}{2}+\mathbb Z$$ $x_0$ belongs to a special
`'symmetric'' nodal line of $v$ and we can apply Proposition 2.4.
Hence we are again back to case 2.
In conclusion, we have shown for case 3, that the touching of a 'normal'
nodal line of an antisymmetric eigenfunction must occur at a point
$x_0\in\mathcal N(w)$. Hence $\mathfrak O(x_0)\subset \mathcal N(u)$
and we are back to case 2.
In the three cases, we have shown that no breaking of canonicity
can occur in contradiction to the assumption that
$\alpha_1(\ell) < \alpha_0(\ell)$.
\begin{remark}\label{past}
We have only used the fact that at $\alpha = \alpha_1(\ell)$ the
considered eigenfunction was canonical for $\alpha<\alpha_1(\ell)$.
\end{remark}
\begin{remark}\label{bound}
The case when $x_0\in \partial \Omega_0$ does not lead to any new problem.
\end{remark}
\subsection{End of the proof of the results for $0<\ell 0$.}\\
Now consider $\mathcal N(u_{\ell})$ and $\mathcal N(u_{\ell'})$
(we supress the $\alpha$) and remember
that we already know that they are both canonical.
Both functions are identically zero on $\tau_0$. \\
We use now the following almost trivial
\begin{lemma}\label{partialintegration}
Let $\varphi_i, i=1,2$ two solutions of~:
\begin{equation}\label{eqf}
H f =\lambda f\;,\text{ in } \Omega_0\;,
\end{equation}
and let $D$ be a simply connected subset of $\Omega_0$ such
that $\varphi_1=0$ for $x\in\partial D$.
We split $\partial D$ in two connected subsets $\partial D_j$ , $j=1,2$ and
assume that $\varphi_2(x)=0$ for $x\in\partial D_1$ and that $\partial D_2$
is a subset of the boundary of a nodal domain of $\varphi_1$. Then either
$\varphi_2(x)=0$ for $x\in
\partial D_2$ or $\varphi_2(x)$ has both signs for $x\in\partial D_2$.
\end{lemma}
{\bf Proof.}
For simplicity we consider just the case of $H=-\Delta + V$.
Then from the fact that, by assumption the functions
$\varphi_j$ solve the same equation (\ref{eqf}), we obtain by Green's
formula~:
\begin{equation}\label{green}
\int_D(\varphi_1\Delta \varphi_2- \varphi_2\Delta \varphi_1)dx=
-\int_{\partial D_2}\varphi_2\frac{\partial \varphi_1}{\partial n}d\sigma =0
\end{equation}
where $\partial/\partial n$ means normal derivative in the outward direction, and $d\sigma$ is the induced measure on $\partial \Omega$.
Now, we observe that, by Hopf's boundary point Lemma and the
assumption on $\varphi_1$, $\frac{\partial \varphi_1}{\partial n}$
keeps the same sign on $\partial D_2$.
Hence as asserted $\varphi_2$ has to have both signs in $\partial D_2$
or vanishes identically there.
\finbox
Using this observation we easily prove the ordering of the eigenvalues.
We take $\varphi_1 = u_{\ell'}$ and
$\varphi_2 = u_\ell$.
Since the zero sets of $\varphi_1$ and $\varphi_2$ are canonical
and $\tau_0$ is in both zero
sets, we just look at a $D$ whose boundary inside $\Omega_0$
is given by $\tau_0$ and a nodal arc
$\sigma_{\varphi_1}$ of $\varphi_1$ such that
\begin{equation}\label{empta}
\mathcal N(\varphi_2)\cap
\sigma_{\varphi_1}=\emptyset\;.
\end{equation}
To see that this is possible we just note that there are $2\ell$ nodal arcs of
$\varphi_2$ and $2\ell'$ nodal arcs of $\varphi_1 $ and that,
due to canonicity, one given nodal
arc of $\varphi_1$ (outside $\partial \Omega_0$) can cross
at most one nodal arc of $\varphi_2$.\\
Now according to the lemma above $\varphi_2$ must vanish at a nodal arc of
$\varphi_1$
which is impossible according to (\ref{empta}).
This is the desired contradiction to the assumption that
$\lambda_\ell =\lambda_{\ell'} $.
The proof that $\text{min }(\lambda_{n/2,s},\lambda_{n/2,a})>\lambda_\ell$ for
$n$ even is identical.\finbox
\textbf{Acknowledgement.}
It is a pleasure to thank A.~Cap, Y.~Chitour and T.~Ramond for their kind
interest. T.~Hoffmann-Ostenhof wants to thank the Department of Mathematics
of the University of Chicago for an invitation where this work was begun.
M. and T.~Hoffmann-Ostenhof thank J.M.~Combes for an invitation to
CPT, Luminy, where this work was continued. B.~Helffer
thanks ESI for an invitation where this work was completed.
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\end{thebibliography}
\scshape
B. Helffer: D\a'epartement de Math\a'ematiques, Batiment 425,
Universit\a'e Paris-Sud, F-91045 Orsay C\a'edex, France.
\slshape
E-mail address: Bernard.Helffer@math.u-psud.fr
\vspace\baselineskip
\scshape
M. Hoffmann-Ostenhof: Institut f\"ur Mathematik, Universit\"at Wien,
\\Strudlhofgasse 4, A-1090 Wien, Austria.
\vspace\baselineskip
\scshape
T. Hoffmann-Ostenhof: Institut f\"ur Theoretische Chemie, Universit\"at
Wien, W\"ahringer Strasse 17, A-1090 Wien, Austria and International Erwin
Schr\"odinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090
Wien, Austria.
\slshape
E-mail address: thoffman@esi.ac.at
\vspace\baselineskip
\scshape
N. Nadirashvili: Department of Mathematics, University of Chicago,
United States.
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