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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 $0<\ell<n/2$  
\begin{equation}\label{mll}
m(\lambda_\ell)=2.
\end{equation}
If $n$ is even, then also
\begin{equation}\label{mn/212}
m(\lambda_{n/2,s})=m(\lambda_{n/2,a})=1.
\end{equation}

\end{theorem}
\begin{theorem}.\label{Thorder}\\
The eigenvalues satisfy the following ordering:\\
When $n$ is odd
\begin{equation}\label{orderodd}
\lambda_{0,s}<\lambda_1<\dots<\lambda_{(n-1)/2}<\lambda_{0,a}\;.
\end{equation}
When $n$ is even
\begin{equation}\label{ordereven}
\lambda_{0,s}<\lambda_1<\dots <\lambda_{n/2-1}<\min (\lambda_{n/2,s},
\lambda_{n/2,a})\le\max(\lambda_{n/2,s},\lambda_{n/2,a})<\lambda_{0,a}\;.
\end{equation}
\end{theorem}
\begin{remarks}.
\begin{enumerate}
\item
Below  we will describe qualitatively the nodal sets
of the groundstate eigenfunctions in the various symmetry subspaces 
$A_{(\cdot)}$ and this will be actually the key 
 point for the proof of the theorems in combination with a comparison 
with  a spherically symmetric  problem
and some perturbation arguments. These arguments apparently do not
carry over in general to the case when we  have just rotational 
symmetry, hence, if the operator just commutes with the actions
of $\mathbb Z_n$.  For $\mathbb D_{2n}$ the simple use of group-theoretical 
arguments seems  hopeless when $n>4$.  
\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 $0<\ell<n/2$ and for $u\in A_\ell$ ($u\neq 0$), $u$ and $gu$ are 
linearly independent. 
\end{lemma}
Note similarly, that we have, for any $k\in \mathbb Z$, 
\begin{equation}\label{gk}
g^{k} \, u =\cos(2\pi k \ell /n) u - \sin (2\pi k \ell /n) v
\end{equation}
and this gives~:
\begin{lemma}\label{Lemma3.2bisb}
For $0<\ell<n/2$ and,  for $u\in A_\ell$  ($u\neq 0$), the 
 real vector space spanned by the $\mathbb Z_n$
 orbit of $u$ (i.e $\mathfrak O_{Z_n}(u)=\{g^ju,j=0,\dots,n-1\}$) 
is a two-dimensional vector space.
\end{lemma}
Using (\ref{gk}) with $k =1$ and $-1$, we get also  an  explicit  formula
 for $v$~:
$$
v = \frac{g^{-1} \, u - g u }{2 \sin (2\pi \ell /n)}
$$ 
Moreover, this defines on $A_\ell$ a map $u\mapsto v = \mathcal R_\ell  u$~:
\begin{equation}\label{2.5}
\mathcal R_\ell :=  \frac{1 }{2 \sin (2\pi \ell /n)} (g^{-1} \,  - g ) \;,
\end{equation}
which according to (\ref{equiv}) is unitary.

\subsection{Further decomposition.}
We have seen that $ T$ exchanges $  B_\ell$ and $ B_{-\ell}$ and that $\mathbb Z_{n}$
 leaves $B_\ell$ invariant.  
We hence can decompose further the space $A_\ell$ into the direct sum~:
\begin{equation}\label{AlSA}
 A_\ell= A_{\ell,s}\oplus  A_{\ell,a},
\end{equation}
where 
\begin{equation}
\begin{array}{ll}
A_{\ell,s}& :=\{ u\in A_\ell\;|\; T u = u\} \\
A_{\ell,a} & :=\{ u\in A_\ell\;|\; T u = - u\}\;.
\end{array}
\end{equation}
We emphasize that these two spaces are no more representation-spaces of the
group, but that it is still meaningful 
 to analyze  the spectrum of the restriction
 of the operator $H$ to  $A_{\ell,s}$ or $ A_{\ell,a}$ which will be denoted 
 respectively by $H_{\ell,s}$ and $H_{\ell,a}$.
 

Moreover, we observe that $\mathcal R_\ell $ anticommutes
 with $T$~:
\begin{equation}
T \mathcal R_\ell = - \mathcal R_\ell T\;.
\end{equation}
So $\mathcal R_\ell $ permits to send isometrically
 $ A_{\ell,s} $ onto $A_{\ell, a} $
 and $ A_{\ell,a} $ onto $A_{\ell, s} $.

As a consequence, we have 
\begin{proposition}\label{PropIsosp}
For $0 < \ell < \frac n2$, the operators $H_{\ell,s}$ and $H_{\ell,a} $
 are isospectral. In particular the multiplicity
 of any eigenvalue $\lambda_\ell^k$
of  $H_\ell$ is even and equals twice
 the multiplicity of $\lambda_\ell^k$ as eigenvalue (in this case denoted by
 $\lambda_{\ell,a}^k$) of 
 $H_{\ell,a}$.
\end{proposition}

This means in particular for showing 
 (\ref{mll}), it suffices to prove  
\begin{equation}\label{multa}
m(\lambda_{\ell,a} ) = 1\text{ for } 0<\ell<n/2.
\end{equation}

\subsection{Lines of symmetry.}\label{ssLin}
We analyze which part of the
 nodal set is determined  directly by symmetry arguments.
We define the nodal set of a solution $u$ of \eqref{divergence} by
\begin{equation*}
\mathcal N(u)=\overline{\{x\in\Omega:u(x)=0\}}
\end{equation*}
A nodal domain of a real valued eigenfunction $u$ is just a component of
$\Omega\setminus \mathcal N(u)$. Nodal sets will be dicussed
further in section 6.
For example, the property that~: $Tu = - u $, implies  that
 $\{x_1=0 \}\cap\Omega$ belongs to the zero set of $u\in A_{\ell,a} $. 
But we can  replace $T$ by another reflection.
The natural question is then to determine,  for which $j$, we have~:
$$ T g^j u = - u\;,\; \forall u\in A_{\ell,a}\;. $$
It is easily seen from (\ref{gk}) that this corresponds
 to the condition~:
\begin{equation*}
\cos (2\pi j \ell / n ) = 1\;,
\end{equation*}
which leads to
\begin{equation}\label{asymlin}
\frac {\ell j}{n} \in \{ 0,\cdots, 2 \ell -1\}\;.
\end{equation}
The corresponding lines of invariance will be called ``special
 antisymmetric lines''. We shall denote the union of these
 special lines
 by~:
\begin{equation}
\mathcal N_{\ell,a} = \{ x\in \Omega\,:\, \omega = \pi \frac jn\text{ for }
j \text { satisfying \eqref{asymlin}}\}.
\end{equation}
Similarily, we can discuss the symmetric case. 
We have  to determine,  for which $j$
$$ T g^j u = - u\;,\; \forall u\in A_{\ell,s}\;. $$
It is easily seen from (\ref{gk}) that this corresponds
 to the condition~:
\begin{equation}
\cos (2\pi j \ell / n ) = -1\;,
\end{equation}
which leads to
\begin{equation}\label{symlin}
\frac {\ell j}{n} - \frac 12 \in \{ 0,\cdots, 2 \ell -1\}\;.
\end{equation}
The corresponding lines of invariance will be called ``special
 symmetric lines''. Note that in the symmetric case we could have
 no special symmetric lines.\\
We  denote the union of these
 special lines
 by~:
\begin{equation}
\mathcal N_{\ell,s} = \{ x\in \Omega\,:\, \omega = \pi \frac jn
\text{ for } j\text{ satisfying \eqref{symlin}}\}.
\end{equation}
So we have proved~:
\begin{proposition}\label{Proposition2.4}
A state  $u\in A_{\ell,a}$  necessarily vanishes on 
$\mathcal N_{\ell,a}$.
Similarily,  $u\in A_{\ell, s }$  necessarily vanishes on
$\mathcal N_{\ell,s}$\;.
\end{proposition}

\subsection { More on the span of an orbit.}
Pick a point $x_0\in \Omega$, then we can look at the orbit of $x_0$
\begin{equation}\label{orbitx0}
\mathfrak O (x_0)=\{hx_0:h\in\mathbb D_{2n}\}.
\end{equation}
There are two cases: 
\begin{equation}\label{x0inT}
\text{ if } x_0 \text{ belongs to  a line of symmetry of $\Omega$, then } 
|\mathfrak O(x_0)|=n
\end{equation}
and otherwise
\begin{equation}\label{x0notinT}
|\mathfrak O(x_0)|=2n.
\end{equation}
The bars denote the cardinality. (Of course for the case of the
disk the origin $\{0\}$ remains invariant under the actions of 
$\mathbb D_{2n}$.)

If $u\in A_\ell$,  it is natural to ask about the linear space generated
by the actions of $ \mathbb D_{2n}$ on $u$, hence the orbit of $u$,
$\mathfrak O(u)$.  
\begin{lemma}\label{spanorbit}
Suppose $0<\ell<n/2$ and  $u\in  A_\ell$. If  
\begin{equation}\label{Tuugu}
Tu\in\text{span }(u,gu), 
\end{equation}
then 
\begin{equation}\label{dimspan}
\text{dim span }\mathfrak O(u)=2,
\end{equation}
otherwise its dimension is four.
In particular the dimension of the span of the orbit of any symmetric or 
antisymmetric function in $ A_\ell$ is two . 
\end{lemma} 
{\bf Proof.}
Suppose first that $Tu\in \text{span }(u,gu)$. This implies also that
$Tg^j u = g^{n-j} T u$ belongs to  $\text{span }(u,gu)$ and hence by Lemma \ref{Lemma3.2bisb} the dimension of the
span of $\mathfrak O(u)$ is two.

If $Tu\not\in \text{ span }(u,gu)$ then also $gTu$ cannot be in this span
and  the dimension of the span must be greater than $3$. On the other hand,  
it has to be even (a multiple of
 the degree of the irreducible representation) and less than four
 using again Lemma \ref{Lemma3.2bisb}.\finbox

\begin{lemma}\label{Lemma2.6}
If $u\in A_\ell$ and if the span of $\mathfrak O(u)$  is two-dimensional, then
there exists $u_{\ell,a}\in A_{\ell,a} $ such that
$$
\mbox{span } \mathfrak O (u_{\ell,a})= \mbox{span } \mathfrak O (u)\;.
$$
\end{lemma}
\begin{lemma}\label{intersection}
Suppose again that $0<\ell <n/2.$  Let $u\in A_\ell$. Then 
\begin{equation}\label{inteNusNua}
 N(g u )\cap \mathcal N(u) \text{ is invariant by the actions of } 
\mathbb Z_n\;.
\end{equation}
\end{lemma}
{\bf Proof.}
This follows from Lemma \ref{Lemma3.2bis}.  
\begin{remark}
In the case when $u$ is antisymmetric (or symmetric), we have the property
 that $\mathcal N(g u )\cap \mathcal N(u)$ is invariant by $\mathbb D_{2n}$.
 Note also, that
\begin{equation}
 \mathcal N(g u )\cap \mathcal N(u) = \mathcal N(\mathcal R  u )
\cap \mathcal N(u) = \mathcal N(w)\;,
\end{equation}
where  $w=u+ i v$ with $v=\mathcal Ru$, (see \eqref{2.5}).
\end{remark} 
\begin{remark}
Without loss  we have chosen a particular reflection axis. 
\end{remark}
\section{Direct spectral consequences of symmetry considerations.} \label{Section3}
First we note that $\lambda_{0,s}$ is the lowest eigenvalue of $H$ which 
is simple and
that the associated eigenfunction  can be chosen to be strictly positive in
$\Omega$. For the eigenvalues $\lambda_{0,a}$ and the 
associated eigenfunctions and for $n$ even
for the eigenvalues $\lambda_{n/2,s},\lambda_{n/2,a}$ and their associated
eigenfunctions  we can use directly the results of the preceeding
section to prove \eqref{mSAs},\eqref{mSAa} and \eqref{mn/212}. We shall 
further show here a part of Theorem 1.3, namely, 
\begin{equation}\label{lambda0a>}
\lambda_{0,a}>\max\{\lambda_{0,s},\max_{0<\ell<n/2}\lambda_\ell\}
\text { and for n
even } \lambda_{0,a}>\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}
$$\lambda_{0,a}>\lambda_\ell\text{  for  } 0<\ell<n/2.$$
\end{lemma}
{\bf Proof.}
Let $\Phi$ be again the restriction of $u_{0,a}$ to $\Omega^0$.
Fix $0<\ell<n/2$ and consider 
\begin{equation}\label{defpsi}
\psi_\ell =\sum_{j=1}^na_j\Phi(g^{j-1}x),\quad\forall x\in\Omega,
\end{equation}
where the $a_j$ are real constants. Note that 
$\Omega^0\cap g^j\Omega^0=\emptyset$ for $0<j<n-1$. We observe that 
$\psi_\ell\in W_0^{1,2}(\Omega)$. We want to pick the $a_j$ such that
$\psi_\ell \in A_\ell$. We just have to take $a_j=\Re e^{2\pi i\ell j/n}$ and 
 define
$$ \psi_\ell (x)=\sum_{j=1}^n\cos(2\pi \ell j/n)\Phi(g^{j-1}x),\quad 
\forall x\in\Omega \;.$$ 
We apply now the variation principle and
get 
\begin{equation*}
\lambda_\ell \le\frac{\langle \psi_\ell \,,\, H\psi_\ell\rangle}{||\psi_\ell||^2}=\lambda_{0,a}.
\end{equation*}
We get strict inequality since by construction the set
$\Omega\setminus\text{supp }\psi_\ell$ has positive measure.
\finbox

With similar arguments we could have also shown that for $n$ even 
$\lambda_{n/2,s}$ and $\lambda_{n/2,a}$ are both strictly greater
than $\lambda_l$ for $0\le \ell<n/2$. But this will be shown in section 9 
in a more direct way.


\section{Reduction to the annulus or the disk and construction
 of an interpolating family.} \label{sRed}


Without changing the divergence structure of $H$, we can reformulate
the eigenvalue problem \eqref{eigenvalue} in such a way, that we can 
replace the operator $H$ with $\mathbb D_{2n}$ symmetry by an operator
still commuting with the actions of $\mathbb D_{2n}$, but $\Omega$ 
becomes $\Omega_0$ which is either the annulus  
$\Omega_0=\{x\in\mathbb R^2:|x|\in (R_1,R_2)\}$
with $0<R_1<R_2$ or the disk with $\Omega_0=\{x\in\mathbb R^2:|x|<R\}$
To do this we observe that by definition there  is a smooth 
diffeormorphism $\Phi$
sending $\Omega$ onto $\Omega_0$, which can be extended from $\overline \Omega$
onto $\overline{\Omega_0}$ and  sending the exterior, respectively the 
interior boundary of $\Omega$  to the corresponding boundary of $\Omega_0$. 
If $\Omega$ is simply connected, $\Phi$ is assumed to map $\Omega$ 
onto the disk. We may also require in the case of $\mathbb D_{2n}$ symmetry 
of our original problem that $\Phi$ commutes with the actions of
$\mathbb D_{2n}$, hence $\Phi \circ h=h \circ \Phi$ for $h\in\mathbb D_{2n}$.

Associated with $\Phi$, there is a canonical unitary tranformation
$\Phi_*$ sending $L^2(\Omega)$ onto $L^2(\Omega_0)$ which is defined by~:
\begin{equation}\label{diffeomorphism}
(\Phi_*u)(y)=J(\Phi)^{1/2}(y)u(\Phi^{-1}(y)),
\end{equation}
where $J(\Phi)$ is the Jacobian of $\Phi$ at $x=\Phi^{-1}(y)$.  
A computation shows that the unitarily equivalent operator
\begin{equation}\label{divergencephi}
H_\Phi=\Phi_*\,H\, \Phi_*^{-1}
\end{equation}
is still of the form as described by \eqref{divergence}, but of 
course for a different matrix $(a_{ij})$. The new potential
$\Phi_*V\Phi_*^{-1}$ turns out to be $V\circ\Phi^{-1}$. Hence it suffices 
to consider \eqref{eigenvalue} on the annulus, respectively on the disk.
By an abuse of notation we will henceforth talk about $H$ instead of $H_\Phi$.
Let $H$ be our operator which commutes with the actions of $\mathbb D_{2n}$
such that $h H=Hh$ for all $h\in\mathbb D_{2n}$
defined on $L^2(\Omega_0)$, $\Omega_0$ the annulus or the disk,
as defined in Section~\ref{Section1}.


We introduce for $\alpha$ in a neighborhood of $[0,1]$ the following 
interpolating  family of operators:
\begin{equation}\label{interpolate}
H(\alpha)=\alpha H +(1-\alpha)(-\Delta)\;,
\end{equation}
whose domain is $W^{2,2}(\Omega_0) \cap W^{1,2}_0(\Omega_0)$.
We consider groundstate  eigenvalues in the 
real symmetry sectors as in Section \ref{Section1} and denote them by
 $\lambda_\ell(\alpha)$, respectively, by $\lambda_{\ell,s}(\alpha)$ or 
$\lambda_{\ell,a}(\alpha)$. 
The corresponding eigenspaces are denoted by $U_\ell(\alpha)$. Note that 
by construction the operator $H_\Phi$ introduced above  
 is in divergence form (see \eqref{divergence}),  
so that the corresponding matrix $a$ 
is strictly positive definite in $\overline\Omega_0$. Hence $H(\alpha)$
is well defined for $\alpha$ in a neighborhood of the interval $[0,1]$, say 
$(-\epsilon_0, 1 + \epsilon_0)$ with $\epsilon_0 >0$ 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 $0<\ell<n/2$~:
$$ I_\ell = \{\alpha\in [0,1+\epsilon_0): m(\lambda_\ell(\alpha))\ge 3\}\;.$$
>From 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 $0<\ell<n/2$.\\
\end{lemma}

\section {Local and global structure of nodal sets.}\label{sNod}

All the material which is presented in this section is rather standard,
 but because we shall need to control the continuity of 
 the nodal sets when varying the parameter $\alpha$, 
it  is helpful
 to recall what is needed in this context. We formulate everything 
for the domain $\Omega_0$ which is either the annulus or the disk.

\subsection{Local structure.}\label{ssnod}
Let $u\in W_0^{1,2}(\Omega_0)$ be a real valued solution of 
\begin{equation}\label{Lu}
Lu=\sum_{i,j}^2a_{i,j}(x)\frac{\partial}{\partial x_i}\frac{\partial}
{\partial x_j}u +\sum_{i=1}^2 b_i(x)\frac{\partial}{\partial x_i} u+ Wu=0,
\end{equation}
where we assume that the matrix $a= (a_{i,j})$ is positive definite with
entries $a_{ij}\in C^\infty(\overline{\Omega_0})$ 
 and that the functions $W$ and  $b_i$ ($i=1,2$) are smooth in
$\overline{\Omega_0}$ and real valued. This implies 
$u\in C^\infty(\overline{\Omega_0})$
For each $x_0\in \overline\Omega_0$, there is a linear non-singular coordinate 
transformation 
$T_{x_0}$ such that  $\widetilde u=u\circ T_{x_0} $ satisfies 
$\widetilde L\widetilde u=0$
 where $\widetilde L$ is of 
the form given in \eqref{Lu} but  with~: $\widetilde a_{ij}(x_0)=\delta_{ij}$.
   
\begin{proposition}{\bf Local behaviour.}\label{Proposition3.1}\\
Let $u\not\equiv 0$ satisfy \eqref{Lu} and $x_0\in {\overline \Omega_0}$. 
Then there exists an $m\in \mathbb N$
 such that~:
\begin{equation}\label{pm}
u(x)=p_m(x-x_0)+\mathcal{O}(|x-x_0|^{m+1})\,,
\end{equation}
where $p_m$ is a homogeneous  polynomial of degree $m$ which
 does not vanish identically and which satisfies the osculating equation
\begin{equation*}
 \sum_{i,j}^2a_{i,j}(x_0)\frac{\partial^2}{\partial x_i\partial x_j}p_m(x-x_0)
=0
\end{equation*}
Moreover, under the  coordinate tranformation, $T_{x_0}$, 
$p_m(x-x_0)$ becomes a harmonic homogeneous polynomial $P_m(x-x_0)$
of degree $m$.

If $x_0\in \partial\Omega_0$ and $u$ satisfies Dirichlet boundary conditions, 
then  
$p_m(x-x_0)$ must have a zero line through $x_0$
which is tangent to $\partial \Omega_0$ at $x_0$. In particular, 
 $m>0$ 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 <n/2\;.
\end{equation}
This follows  from the considerations which lead to  Lemma \ref{Lemma3.2bis}.

Note that in polar coordinates the harmonic homogeneous polynomials
$P_m$ have the simple representation:
\begin{equation*}
P_m(r,\omega)=r^m(a\cos m\omega+b\sin m\omega).
\end{equation*}
{\bf Proof of Proposition \ref{Proposition3.1}.}
This follows for our case  just from a Taylor 
expansion with remainder (see \cite{Bers:1955}). 
 \finbox 

Recall that the  nodal set  
$\mathcal {N}(u)$ is defined by  
\begin {equation*}
\mathcal {N}(u)=\overline{\{x\in  \Omega_0 : u(x)=0\}}
\end {equation*}
and a nodal domain of $u$ is a connected component of
 $\Omega\backslash \mathcal {N}(u)$.
\begin{proposition}\label{Proposition1.1} (see e.g. \cite {HoMiNa:1999}) \\
(i) $\mathcal N(u)$ is a union of smoothly immersed circles in $\Omega_0$ 
and immersed arcs connecting points of $\partial\Omega_0$. Each of these 
is called a {\em nodal line}. Note that self-intersections are 
allowed. Maximal embedded pieces of nodal lines will be called
{\em nodal arcs.} In addition intersections of these immersed arcs, 
circles and self-intersections are transversal. The immersed arcs hit 
$\partial\Omega_0$ transversally.\\
(ii) If for some $x\in\Omega_0$, $u(x)=0$,  but 
$(\nabla u)(x)\ne 0$, then  
$x$ lies on exactly one nodal  line and is not a point of self-intersection.
On the other hand if $x_0\in \Omega_0$ is not a point of self-intersection 
and lies on a single nodal line of $u$, then
$(\nabla u)(x_0)\neq 0$. Also if 
$x_0\in \partial\Omega_0\setminus \mathcal N(u)$, 
then also $(\nabla u)(x_0)\neq 0$. \\
(iii) If for some $x\in \Omega_0$  and for some $p\in\mathbb N$,
$u(x)=0, (\partial^{\beta}u)(x)=0 \text{ for } |\beta|\le p$
but $(\partial^{\alpha}u)(x)\ne 0$  for some multiindex $\alpha $ 
of length $p+1$, then exactly 
$p +1$ distinct nodal lines go  through $x$ whose tangents meet
transversally at $x$. \\
(iv) If $x$ is a zero of  $u$ as in (iii) but lies in $\partial \Omega_0$,
then the same property is  true with the convention 
that one of the nodal lines  is a component of  $\partial\Omega_0$. 
\end{proposition}
The proof of these results is given in \cite{HoMiNa:1999} 
and follows also partially 
from  Proposition \ref{Proposition3.1}.



Following \cite{HoMiNa:1999} we can introduce some definitions:
\begin{definition}\label{Definition1.1}
An {\em abstract nodal set} is a set $\mathcal N\subset\overline\Omega_0$ 
satisfying  
 \ref{Proposition1.1}(i), where we do not 
require that it is 
actually the nodal set of an eigenfunction.\\
An {\em isotopy} of nodal sets is a curve of nodal sets such that each 
immersed circle or arc moves along a smooth isotopy which respects 
nodal arcs. \\
A {\em nodal pattern} is an isotopy class of nodal sets.
\end{definition}

\section{The radial case and canonicity.}
\label{sCan}
\subsection{Radial case.}\label{ssRad}
We first analyze the radial case. Here we
take $\Omega_0$ to be the annulus or the disk, 
and $H=-\Delta$ with Dirichlet boundary conditions. \\
By a standard reduction to ordinary differential equations,
we find that any $u_\ell^k\in  A_\ell$ which solves the equation 
$-\Delta u_\ell^k=\lambda_\ell^ku_\ell^k$, can be written in 
polar coordinates as 
\begin{equation}\label{nod}
u_\ell^k=f_\ell^k(r)(a\sin \ell\omega+b\cos \ell \omega)
\end{equation}
where $(a,b)\ne (0,0)$ and $f_\ell^k$ satisfies for $\ell\in \mathbb N$
\begin{equation*}
(-d^2/dr^2-r^{-1}d/dr+r^{-2}\ell^2 )f_\ell^k(r )=\lambda_\ell^kf_\ell^k
\end{equation*}
with $f_\ell^k(R_i)=0$, for $i=1,2$.\\
 If we split $ A_\ell$ into
$ A_{\ell,s}$ and $ A_{\ell,a}$,  then  
the cosine part, respectively the sine part in \eqref{nod}, corresponds
to $ A_{\ell,s}$, respectively $ A_{\ell,a}$.  
(The $f_\ell ^k$ are of course
Bessel functions, but we will not use this.) \\ 
Taking $k=1$ and writing for simplicity 
$\lambda_\ell=\lambda_\ell^1,u_\ell =u_\ell^1$,
 these observations imply that
\begin{equation}
m(\lambda_\ell)=2, \text { for } \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 $0<\ell <n/2$ and define as in section 3
\begin{equation}\label{tauj}
\tau_j=\{x\in\overline{\Omega_0}:\omega=\pi j /n\},\quad j=0,\dots,2n-1\;,
\end{equation}
and  
\begin{equation}\label{Omega0j}
\Omega_{0j}=\{x\in\overline {\Omega_0}:\omega\in(\pi j/n,\pi(j+1)/n)\},
\quad j=0,\dots,2n-1\;.
\end{equation}
The $\tau_j,\tau_{j+1}$ are just the boundaries of 
$\Omega_{0j}$ without  $\partial\Omega_0$.

The nodal set of a $\phi_{\ell,rad}=a\,u_{\ell,rad} + b\, v_{\ell,rad} =
f_\ell(r)(a\cos \ell\omega+b\sin \ell \omega)$. 
consists of $2\ell $ components given by
\begin{equation} \label{Nradial}
\mathcal N(\phi_{\ell,rad})={\{x\in\overline{\Omega_0}\,:\, a\sin \ell \omega+
b \cos \ell \omega=0\}}
\end{equation}
Thereby we assume without loss that $f_\ell(r)>0$ 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 $0<\ell<n/2$
\begin{equation}\label{Ncano}
 u_\ell \quad \text{ is {\bf  canonical.}}
\end{equation}
For $n$ even  
\begin{equation}\label{Nn/2}
\mathcal N(u_{n/2,s})=\cup_{j=0}^{n-1}\tau_{2j} \text{ and }
\mathcal N(u_{n/2,a})=\cup_{j=0}^{n-1}\tau_{2j+1}.
\end{equation} 
\end{theorem}

\begin{remarks}.
\begin{enumerate}
\item \eqref{NuS,As},\eqref{NuS,Aa} as well as \eqref{Nn/2} have been shown
in section \ref{Section3}.
\item
We could have defined in an analogous way also the canonicity 
of any function in the $\mbox{span }\mathfrak O( u_\ell)$ which would amount
to saying that all $u\in\mbox{span }\mathfrak O(u_\ell)$ have the same 
nodal pattern and some additional localization of the nodal lines.
 But for the proof of our results we just need the canonicity
of $u_\ell$. 
\end{enumerate}
\end{remarks}


%% Finally, if $\lambda_{n/2,1}=\lambda_{n/2,2}$ the associated 2-dimensional
%% eigenspace $U_{n/2}$ is {\bf canonical}, i.e 
%% \begin{equation}\label{Nn/2cap}
%% \mathcal N(u_{n/2,1})\cap\mathcal N(u_{n/2,2})=\emptyset.
%% \end{equation}


\section{Stability of the canonicity.}\label{sStab}
The next lemma will show the stability of canonicity with 
respect to the parameter $\alpha$. 
\begin{lemma}\label{Lemma5.4}
Let $\alpha_0(\ell)$ be given according to \eqref{alpha0}.
The set of $\alpha$'s in $(-\epsilon_0, \alpha_0(\ell))$, 
such that $u_\ell(\alpha)$ is canonical is open.
\end{lemma}
Since  $\dim U_{\ell,a}(\alpha)=1$, $u_\ell(\alpha)$
 is well defined up to multiplication by a scalar.\\
$u_\ell(0)=u_{\ell,rad}$ is 
canonical, see section 7.2, and this implies directly
\begin{lemma}\label{Lemma5.3}
For any $0<\ell < n/2$, there exists an ${\tilde \alpha_1}(\ell) >0$ 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 $0<\ell<n/2$. Then 
\begin{equation}\label{al01}
\alpha_1(\ell)=\alpha_0(\ell).
\end{equation}
\end{proposition}
{\bf Proof.}
For the proof  we shall investigate how a nodal line, which is not a special
antisymmetric line, see section 2.3, can be deformed in a continuous 
way such that the canonicity is violated.

{\bf Assume for contradiction $\alpha_1(\ell)<\alpha_0(\ell)$.} \\

>From 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 $0<\ell <n/2$.}
In order to conclude the proofs of \eqref{mll} and \eqref{Ncano}, 
it remains to prove 
\begin{lemma}
$$ \alpha_0(\ell) = 1 +\epsilon_0\;. $$
\end{lemma}
We  proceed via contradiction and assume that~:
$$
 \alpha_0(\ell) < 1+\epsilon_0\;.
$$
According to Lemma 5.1, we have 
\begin{equation}
m\left(\lambda_{\ell,a}(\alpha_0(\ell))\right)=2 \;,
\end{equation}
and 
\begin{equation}\label{UA12}
U=U^1\oplus U^2=U_{\ell,a}(\alpha_0(\ell))=U^1_{\ell,a}(\alpha_0(\ell))
\oplus U^2_{\ell,a}(\alpha_0(\ell))
\end{equation}
where $U^1$ is one-dimensional.

We now introduce~:
\begin{equation}
u=\lim_{\alpha \uparrow  \alpha_0(\ell)} u_{\ell}(\alpha)\;.
\end{equation}

We first  show that  Proposition \ref{alpha1=0}
 implies also that $u$ is canonical as a continuous limit
 of a family of  antisymmetric canonical eigenfunctions.

 If $U$ has dimension $\ge 2 $, it is immediate to find 
a non-canonical $\phi  \in U$. Pick just  $f\in U^2$ ($f\neq 0$) and take 
$x_0\in \Omega_{00}$.
 Then,  for some $(a,b)\neq(0,0)$, $\phi (x_0)=(a\, u  +b\, f)(x_0)=0$.
This contradicts $\bf C3(\ell)$.\\
We can now introduce, for $\theta\in [0,\frac \pi 2]$~:
\begin{equation}\label{wtheta}
w_\theta=\cos\theta \,  u + \sin \theta \, \phi\;.
\end{equation}
Let $\theta_0$ be the smallest positive 
$\theta<\frac \pi 2$ such that
$w_{\theta_0}$ is not canonical. 
Again, $w_{\theta_0}$ appears as a limit 
(as $\theta \uparrow  \theta_0$)  of canonical
 antisymmetric states and we get a contradiction as above.


Proposition 9.2 and Lemma 9.5 imply that $m(\lambda_\ell)=2$ and that 
$u_\ell$ is canonical  
completing the proofs of Theorem 1.2 and Theorem 7.2.
\finbox
 

\section{Proof of Theorem 1.3.}
Part of Theorem 1.3 has been already shown in section 3.
We still  have to show the ordering of the $\lambda_\ell$ for $0<\ell<n/2$ and 
the inequality between these $\lambda_\ell$ and $\lambda_{n/2,s}$ and 
$\lambda_{n/2,a}$ for $n$ even.
We consider first
the inequalities  amongst the $\lambda_\ell$ for  $0< \ell <n/2$. 
Hence we take $0<\ell<\ell'<n/2$ and consider the
corresponding antisymmetric groundstate eigenfunctions 
$u_\ell$ and $u_{\ell'}$. We know that for $\alpha=0$ the ordering holds
and hence by section 5 also for sufficiently small positive $\alpha$.

{\bf We again proceed via contradiction and assume 
that $\lambda_\ell(\alpha)=\lambda_{\ell'}(\alpha)$ for some $\alpha>0$.}\\
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.

 
\footnotesize
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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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