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Multiparticle Schr\"odinger operators, homogeneous magnetic field, 
pseudomomentum.
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\Volume{248}
\Copyrightissue{248}
\Month{01}
\Year{2003}
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\Receiveddate{10 February 2003}
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\keywords{Multiparticle Schr\"odinger operators, homogeneous magnetic 
field, pseudomomentum.}
\subjclass[msc2000]{81Q10}
\title[Bound States of Atoms]{Bound States of Atoms in a Homogeneous 
Magnetic Field.}
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\author[S. Vugalter]{Semjon Vugalter\footnote{Corresponding
      author: e-mail: {\sf wugalter@mathematik.uni-muenchen.de}, 
Phone: +49\,089\,2180\,4405
      }\inst{1}} \address[\inst{1}]{Mathematik, Universit\"at 
M\"unchen, Theresienstrasse 39, 80333
		M\"unchen, Germany}


\begin{abstract}
We consider Schr\"odinger operators of atoms in a homogeneous 
magnetic field. 
We prove that for each fixed value of the 
pseudomomentum, the corresponding operator has an infinite number of 
eigenvalues. The asymptotic of the counting function is studied for 
values of spectral parameter near the bottom of the essential 
spectrum.
\end{abstract}
\maketitle

\section { Introduction. }

A fundamental result in the theory of multiparticle Schr\"odinger 
operators concerns the existence of ground states for atoms and 
positive ions. The existence of a ground state is expressed by the 
fact that after separation of the center of mass motion, the reduced 
Hamiltonian has a nonempty discrete spectrum. Moreover, it is known 
that the number of discrete eigenvalues is infinite (Zhislin's 
theorem~\cite{RS}), and the asymptotics of the counting function was 
obtained in~\cite{VZh1}.

The goal of the paper at hand is to generalize these results to the 
case of atoms in a homogeneous magnetic field. The first and the main 
obstacle on this way is the separation of the center of mass motion. 
In a homogeneous magnetic field, it can only be performed in the 
direction of the field, but not in the plane orthogonal to it. A way 
to overcome this obstacle was found by J.~Avron, I.~Herbst and 
B.~Simon in~\cite{AHS}. They observed that the Hamiltonian commutes 
with the pseudomomentum operator, and that instead of separating off 
the center of mass motion, one can fix the values of the 
pseudomomentum. For the reduced Hamiltonian (operator with fixed 
pseudomomentum), J.~Avron, I.~Herbst and B.~Simon proved a theorem of 
HVZ type. They proved that the bottom of the essential spectrum of 
the reduced Hamiltonian for an atom coincides with the infimum of the 
spectrum of the   operator obtained by discarding the interactions 
between one of the electrons with the rest of the system.

Although this result shows that the discrete spectrum of the reduced 
Hamiltonian corresponds to bound states of an atom in a homogeneous 
magnetic field, and gives a variational characterization of bound 
states, it is actually not possible to apply it to prove the 
existence of bound states. Without magnetic field, the bottom of the 
essential spectrum for the Hamiltonian of an atom corresponds to the 
infimum of the spectrum of the corresponding operator (after 
separation of the center of mass motion) obtained by removing one 
electron. This Hamiltonian always has an eigenvalue at the bottom of 
its spectrum, and its discreteness plays a very important role both 
in the proof of Zhislin's theorem, and in computing spectral 
asymptotics. In contrast, the operator that defines the bottom of the 
essential spectrum of the reduced Hamiltonian according to~\cite{AHS} 
possesses only essential spectrum, and can hardly be used for 
constructing variational trial functions to prove the corresponding 
analogue of Zhislin's theorem.
This circumstance, combined with the complicated form of operators with fixed pseudomomentum, contributed to a long-time lack of progress in the study of the bound states of atoms in a homogeneous magnetic field.

An attempt to find a different solution, subsituting the method of separating the center of mass motion, was carried out by G.~Zhislin and the author in~\cite{VZh4, V}, where fixation of the type $m$ of the rotational $SO(2)$ symmetry was used. It was proved that for systems with total charge $Q$ nonequal to zero, the HVZ theorem holds on subspaces with fixed $m$. This result was used in~\cite{V} to study the discrete spectrum of molecular ions. However, a counterexample shows that for neutral systems in a homogeneous magnetic field (and atoms are neutral), fixation  of $m$ can not be a substitute for the separation of center of mass motion.

A breakthrough in the study of the discrete spectrum of atomic Hamiltonians with fixed pseudomomentum was achived in~\cite{VZh}, where a Hydrogen type system was studied. The cluster operator, used in~\cite{AHS} was modified in~\cite{VZh} in such a way that its lowest spectral point, which defines the bottom of the essential spectrum of the whole operator, turned (in case of Hydrogen) into a discrete eigenvalue. It allowed, in the case of Hydrogen, to prove existence of bound states for all values of the pseudomomentum, and to find the spectral asymptotics. The results of~\cite{VZh} were generalized by G.~Zhislin in~\cite{Zh} to the case of Helium. In both cases~\cite{VZh} and~\cite{Zh}, the proofs use the fact that any cluster in a nontrivial cluster decompositions can not contain more than one electron. Atoms with more than two electrons do not have this property.

In the present paper, we prove the existence of discrete eigenvalues for an atom with three electrons for fixed pseudomomentum, and the method of the proof can be straightforwardly generalized to the case of atoms with an arbitrary number of electron. We also show that the number of eigenvalues ${\cal N}(\lambda )$, counted with their multiplicities, which have a distance to the bottom of the essential spectrum greater than $\lambda$, tends to $\infty$ as $\lambda ^{-\frac1{2}}$ in the limit $\lambda \to +0$. We recall that for atoms without magnetic field, ${\cal N}(\lambda )\to \infty $ as $\lambda ^{-3/2}$. In this sense, an atom, coupled to a homogeneous magnetic field at fixed pseudomomentum, is similar to a system of one dimensional particles. This phenomenon was earlier observed in the cases of Hydrogen and Helium~\cite{VZh,Zh}, and in the case where the magnetic field becomes infinite at spatial infinity~\cite{VZh3}. The central part of the proof in the present paper is the construction of the cluster Hamiltonian $\tilde H_{\nu}^C(B)$, the lowest discrete eigenvalue of which coincides with the bottom of the essential spectrum of the reduced Hamiltonian $H_{\nu}(B)$. Under the assumption, that the cluster operator with the properties described above  can be constructed, the asymptotic of the counting function for an arbitrary atom was found earlier ~\cite{Zh}.
 The difference between the cluster operator  $\tilde H_{\nu}^C(B)$, and the one invented in~\cite{AHS} is the following. In the operator $\tilde H_{\nu}^C(B)$, the center of mass motion in the direction of the field is separated also for the subsystem $C$ with $N-1$ electrons. The proof that such a separation is sufficient to ensure the discreteness of the lowest spectral point of $\tilde H_{\nu}^C(B)$ is based on the following observation. All particles except one in the system are identical (electrons). For identical particles, the center of mass motion can  be separated and the kinetic energy operator of the motion of the electrons can be written as a direct sum of two operators. One of them correspons to the relative motion of the electrons. The second one is the kinetic energy operator of an effective particle with the charge and
 the mass $N$ times greater than the charge and the mass of an electron. On the other hand, the pseudomomentum operator depends only on the coordinates of the nucleus, and the center of mass of the electrons. As a result, the kinetic energy can in the atomic case be written as a direct summ of two operators. The first operator is the same as for the Hydrogen atom with a fixed pseudomomentum, but with the electron replaced by the effective particle. The second operator depends on relative coordinates of the electrons and does not depend on the pseudomomentum.

For Hydrogen, it was proved in~\cite{VZh} that states supported in regions, where the distance to the origin on the plane orthogonal to the magnetic field is large, can not have low energy. In the multi-electron case, it results in energetically disadvantageous states with at least one electron far from the origin on the plane orthogonal to the field. Ionization in the direction of magnetic field requires less energy. 

To prove the existence of bound states for the cluster Hamiltonian (as well as for the full reduced Hamiltonian) we have mainly to consider possibilities of motion of the electrons in the direction of the field. The kinetic energy of this motion has standard form (except the dimension of the space) and presence of long-range attractive effective interaction ensures existence of discrete eigenvalues on the same way as for atoms without magnetic field.   
 

After the results of the work at hand were annonced and discussed with 
G.Zhislin, he gave  a different proof of the main theorem, which can be 
found in~\cite{Zh1}.



\section { Main Definitions and Results. }
Let
\begin{equation}
{\cal H}(B) = M_0^{-1}(i\nabla _0 +A_0)^2 + 
\sum\limits_{j=1}^N(i\nabla _j +A_j)^2 -
\sum\limits_{j=1}^N N|r_j -r_0|^{-1} + \sum\limits_{s,t = 1 \atop 
{s<t}}^N|r_s -r_t|^{-1}\ ,
\end{equation}
$$
r_j = (x_j, y_j, z_y), \quad A_0= \frac{B}2 \{ -Ny_0, Nx_0, 0\}, 
\quad A_j= \frac{B}2 \{ y_j, -x_j, 0\} \quad   j\ge 1
$$
be the Schr\"odinger operator of an $N$-electron  atom in a 
homogeneous magnetic field $B$. We assume that the particle with the 
number $0$ is the nucleus, the charge of an electron is $-1$, the mass 
of an electron is $\frac1{2}$ and the nuclear mass is $\frac1{2}M_0$. 
The direction of the magnetic field coincides with the direction of 
$z$-axis. Operator  ${\cal H}(B)$ acts in ${\cal L}_2({\mathbb 
R}^{3(N+1)})$. The operator ${\cal H}(B)$ - is invariant under 
translations in the direction of the magnetic field, and the center 
of mass motion in this direction can be separated on a standard way. 
To this end, we use the notation
\begin{equation}
M= M_0 + N ,\qquad \zeta _z = M^{-1}\{M_0 z_0 + \sum\limits_{j=1}^N z_j\}
\end{equation}
and define a space
\begin{equation}
R_{0z}= \{ r|r \in {\mathbb R}^{3(N+1)}\ ,\ M_0z_0 + 
\sum\limits_{j=1}^N z_j=0\}
\end{equation}
with the scalar product
\begin{equation}
(r,\ \tilde r)_1 = z_0 \tilde z_0M_0 + \sum\limits_{j=1}^N z_j \ 
\tilde z_j + \sum\limits_{j=0}^N \{x_j\tilde x_j + y_j \tilde y_j\} \ 
.
\end{equation}
The space $R_{0z}$ can be written as a tensor product of the spaces 
$R_{0z}^1$ and ${\mathbb R}^{2(N+1)}$, where
\begin{equation}
R_{0z}^1 = \{ z | z =(z_0, \ldots , z_N), \quad M_0z_0 + 
\sum\limits_{j=1}^N z_j= 0\}\ .
\end{equation}
Let $\tilde{\Delta}_{0z}$ be the Laplacian on $R_{0z}^1 $,
\begin{equation}
{\Delta}_{0z} = \tilde{\Delta}_{0z}\otimes I({\mathbb R}^{2(N+1)})\ .
\end{equation}
Operator ${\cal H}(B)$ is a direct summ of two operators
\begin{equation}
{\cal H}(B)= H(B)\otimes I_1 + I_{3N+2}\otimes 
\Bigl(-M^{-1}\frac{\partial ^2}{\partial \zeta _z}\Bigr)\ ,
\end{equation}
where $H(B)$ is the Schr\"odinger operator after  separating  the 
center of mass motion in the direction of the $z$ - axis, 
$-M^{-1}\frac{\partial ^2}{\partial \zeta _z^2}$ - the kinetic energy 
of the center of mass motion in this direction, $I_1$ and $I_{3N+2} $ 
are the identity operators on ${\mathbb R}^1$ and $R_{0z}$ 
respectively.

Operator $H(B)$ is given by the equality:
\begin{equation}
H(B)= T_{\perp} - \Delta _{0z}+V\ ,
\end{equation}
where
\begin{equation}
T_{\perp} = M_0^{-1}(i\nabla _{0\perp } + A_{0\perp })^2 + 
\sum\limits_{j=1}^N(i\nabla _{j\perp } + A_{j\perp })^2\ ,
\end{equation}
\begin{equation}
V = -\sum\limits_{j=1}^N N |r_j - r_0|^{-1} + \sum\limits_{s,t=1 
\atop {s<t}}|r_s - r_t|^{-1}\ .
\end{equation}
Here and everywhere below,  the index $\perp$ marks projections of 
different vectors onto the plane orthogonal to the field. In 
particular
$\nabla _{i\perp }$, $A_{i\perp }$ $\ i=0, \ldots , N$ - the 
projections of  $\nabla _i$ and $A_i$ onto the plane orthogonal to 
the direction of the field.

Our first goal is to find a suitable operator, whose discrete 
eigenvalues correspond to the bound states of the system. In other 
words to find an operator for which an HVZ-type theorem holds. An 
operator posessing this property was found by J. Avron, I. Herbst and 
B. Simon~\cite{AHS}.

Let
\begin{equation}
{\cal P}_\perp = \sum\limits_{j=0}^N(i\nabla _{j\perp } - A_{j\perp })
\end{equation}
denote the operator for the pseudomomentum. As it was observed in~\cite{AHS}, 
the operator ${\cal P}_{\perp}$ commutes with $H(B)$ and in the atomic case 
(neutral systems), the two components of ${\cal P}_{\perp}$ commute. 
The operator $H(B)$ can be written as a direct integral of operators 
$H_\nu (B)$ acting on fiber-spaces of the states with fixed values 
$\nu = (\nu _1,\ \nu _2)$ of pseudomomentum
\begin{equation}
H(B)=\int\limits_{{\mathbb R}^2}^{\oplus}H_{\nu} (B)d\nu \ .
\end{equation}
The explicit formula for the reduced Hamiltonian $H_{\nu}(B)$ can be 
found in~\cite{VZh}. To avoid long notations we will give it only for 
an atom with three electrons in section 5.
As it was proved in~\cite{AHS} for operators $H_{\nu}$ the HVZ 
theorem holds. The discrete spectrum of operators $H_{\nu}$ 
corresponds to bound states of atoms.

Let $\mu _{\nu}= \inf \sigma _{ess}\{H_{\nu}\}$, $\ {\cal N}(\mu 
_{\nu}- \lambda )$ be the number of eigenvalues of $H_{\nu}$, counted 
with their multiplicity, which are less than $(\mu _{\nu} - \lambda 
)$.
The main result of the present paper is the following:
\begin{thm}
For all values of  $\nu = (\nu _1,\ \nu _2) $ $\sigma 
_{disc}(H_{\nu})\ne \emptyset $. Moreover, for all $\nu$, the operator 
$H_\nu$   has infinite number of discrete eigenvalues, and
\begin{equation}
{\cal N}(\mu _{\nu} - \lambda )\cdot \lambda ^{1/2}\to {\rm const \ 
as\  } \lambda \to +0 \ .\label{4.1}
\end{equation}
The constant on the r.h.s. of~{\ref{4.1}} does not depend on $\nu$.
\end{thm}

{\em Remarks.}
\begin{enumerate}
\item[(1)]
Notice that without magnetic field, the number of eigenvalues ${\cal 
N}(\mu - \lambda )$ of the operator with fixed total momentum goes to 
$\infty$ as $\lambda ^{-3/2}$ in contrast to $\lambda ^{-1/2}$ in 
Theorem 2.1. From this point of view a system in a homogeneous 
magnetic field with fixed pseudomomentum is similar not to a two 
particle Coulomb system in dimention three, but to a one-dimentional 
system. This phenomenon will be discussed in section 4.
\item[(2)]
The results of Theorem 2.1 can be easily  generalized to the case of 
subspaces of fixed permutational symmetry similar to~\cite{VZh1, Zh, 
VZh2}. In particular, Theorem 2.1 holds if the Pauli principle is 
taken into account. Notice, that the asymptotics of the counting function with symmetry for atoms with arbitrary number of electrons were obtained earlier in  ~\cite{Zh} under the additional assumption, that a statement similar to our Theorem 3.1 (see section 3) holds. The main achievement of the work at hand is the proof, that this assumption for all atoms really holds.
\item[(3)]
The precise value of the constant on the r.h.s. of ~{\ref{4.1}} can 
be found in ~\cite{Zh}.
\end{enumerate}

\section { Reformulation of the HVZ Theorem. }
Let
$C = (0,\ 1,\ \ldots \ ,\ N-1)$ ,
$T_{\perp \nu }$ - is the operator $T_\perp $ restricted onto the 
states with pseudomomentum $\nu$ and
$$
H_{\nu }^C=T_{\perp \nu }-\Delta _{0z} + V^C\ ,
$$
where
\begin{equation}
V^C = -\sum\limits_{j=1}^{N-1}N|r_j - r_0|^{-1} +\sum\limits_{s,t=1 
\atop {s<t}}^{N-1}|r_s - r_t|^{-1}\ .
\end{equation}
As it follows from the results of~\cite{AHS}
\begin{equation}
\mu _{\nu} = \inf \sigma _{ess} (H_{\nu}(B))= \inf \sigma (H_{\nu 
}^C(B))\ .\label{7.2}
\end{equation}
Although equality~(\ref{7.2}) generalizes the HVZ theorem for atoms 
without magnetic field to the case of atoms in a homogeneous magnetic 
field, it is not suitable for the proof of Theorem 2.1. A variational 
proof of the infiniteness of the number of discrete eigenvalues of 
atoms without magnetic field (Zhislin theorem~\cite{RS} ) is based on 
the fact that the infimum of the essential spectrum of an atom 
coinsides with the lowest eigenvalue of the same atom without one 
electron (after separation of its center of mass motion). The 
eigenfunctions corresponding to this eigenvalue play an important role 
in construction of the trial functions for the whole operator. In 
contrast to the case $B=0$, the operator $H_{\nu}^C(B)$ for $B\ne 0$ 
has only essntial spectrum. To prove Theorem 2.1 we need to 
reformulate the HVZ theorem for the operator $H_{\nu}(B)$ and to find an 
operator different from $H_{\nu}^C(B)$ , for which $\mu _{\nu}$ is 
the lowest eigenvalue.

To this end, let us define an operator $\Delta _{0z}[C]$ similar to 
$\Delta _{0z}$ , but with the full system replaced by the subsystem 
$C = (0,\ \ldots \ ,\ N-1)$ and let
\begin{equation}
\tilde H_{\nu}^C(B)= T_{\perp \nu} - \Delta _{0z}[C]+V^C\ . \label{8.1}
\end{equation}
The following theorem plays the central role in the proof of Theorem 2.1:
\begin{thm}
Operator $\tilde H_{\nu}^C(B)$ has the following properties:
%\begin{enumerate}
%item[i)]
%$\mu _{\nu} = \inf \sigma _{ess}( H_{\nu}(B))= \inf \sigma (\tilde 
%H_{\nu}^C(B))\ ;$
%\item[ii)]
%$\mu _{\nu} \in \sigma _{disc}(\tilde H_{\nu}^C(B))\ .$
%\end{enumerate}
\begin{eqnarray*}
& i) & \mu _{\nu} = \inf \sigma _{ess}( H_{\nu}(B))= \inf \sigma 
(\tilde H_{\nu}^C(B))\ ;\\
& ii) & \mu _{\nu} \in \sigma _{disc}(\tilde H_{\nu}^C(B))\ .
\end{eqnarray*}
\end{thm}

\section { Proof of Theorem 2.1. }
Let
\begin{equation}
\Theta = (\Theta _1,\ \Theta _2,\Theta _3)= (M_0r_0 + 
\sum\limits_{j=1}^{N-1}r_j)\cdot (M_0 + N-1)^{-1}-r_N\ .
\end{equation}
Operator $H_\nu (B)$ can be written as
\begin{equation}
H_\nu (B) = \tilde H_{\nu}^C(B)- \frac{M_0+N}{M_0+N-1}\ 
\frac{\partial ^2}{\partial \Theta _3^2} - N|r_0 - r_N |^{-1} + 
\sum\limits_{i=1}^{N-1}|r_i - r_N|^{-1}\ .\label{9.2}
\end{equation}
We recall, that without magnetic field $(B=0)$, after the separation of 
mass motion, the Schr\"odinger operator of the same atom can be 
written as
\begin{equation}
  H_0 = H_0^C - \frac{M_0+N}{M_0+N-1} \Delta _{\Theta} - N|r_0 - r_N 
|^{-1} + \sum\limits_{i=1}^{N-1}|r_i - r_N|^{-1}\ , \label{9.3}
\end{equation}
where $H_0^C$ is the Hamiltonian of the subsystem $C=(0,\ \ldots ,\ N-1)$.

For both operators $H_{\nu}(B)$ and $H_0$ the lowest point of the 
essential spectrum is an eigenvalue of $\tilde H_{\nu}^C(B)$ or 
$H_0^C$ respectively. The only one important difference 
between~\ref{9.2} and~\ref{9.3} is that in the operator $H_{\nu}(B)$, the 
three-dimentional operator $\Delta _{\Theta}$ replaced by the
one-dimentional $\frac{\partial ^2}{\partial \Theta ^2_3}$.
As a result, Theorem 2.1 can be proved analogously to the theorems 
for operator $H_0$ in~\cite{VZh1} if we replace the eigenfunctions of 
the operator
\begin{equation}
  -\frac{M_0+N}{M_0+ N-1}\ \Delta _{\Theta} - \frac1{|\Theta |}
\end{equation}
in the region $|\Theta |>C_0$ with  Dirichlet boundary conditions 
at $|\Theta |=C_0$ by the eigenfunctions of the corresponding 
one-dimentional operator
\begin{equation}
-\frac{M_0+N}{M_0+N-1}\ \frac{\partial ^2}{\partial \Theta 
^2_3}-\frac1{|\Theta _3|}
\end{equation}
in the regions $\Theta _3 > C_0$ and $\Theta _3 < -C_0$ with 
Dirichlet boundary condition at $\Theta _{3} = {+ \atop {(-)}}C_0$ 
 In the estimate of the upper bund for the number of eigenvalues, we 
replace the standard partition of unity of the configuration space by a 
partition depending on $z$-coordinates of particles only. Such a
variant of partition of the unity applied to the configuration space has been 
used several times in the past (see~\cite{VZh,Zh,VZh3}). In the present 
paper we will not repeat estimates published earlier and will focus 
 on the main new part -the  proof of Theorem 3.1.

\section { Proof of Theorem 3.1. }
To simplify the notations we will prove the theorem only in the case 
$N=3$ . A generalization to the case $N>3$ is staightforward. For the 
subsystem consisting of three electrons, we can introduce Jacobi 
coordinates.
Let
\begin{eqnarray}
r_{C\perp} : & =  (x_C,\ y_C) & = 
\frac1{3}\sum\limits_{j=1}^3r_{j\perp};\quad q_\perp : = (x_q,\ y_q) 
= r_{2\perp}-r_{3\perp}\ , \nonumber\\
\zeta _{\perp} : & =  (x_{\zeta}, \ y_{\zeta}) & = 
\frac1{2}(r_{2\perp}+r_{3\perp})-r_{1\perp}\ .
\end{eqnarray}
In these coordinates, the operator $T_\perp$ can be written as
\begin{equation}
T_\perp = M_0^{-1}(i\nabla _{0\perp }+A_{0\perp})^2 + \frac1{3} 
(i\nabla _{r_{C\perp}}+ A_{C\perp})^2 + (i\nabla 
_{q_{\perp}}+A_{q_{\perp}})^2 + (i\nabla _{\zeta _{\perp}}+ 
A_{\zeta_{\perp}})^2\ , \label{11.2}
\end{equation}
where
$$
A_{C\perp}= 3B\{y_C,\ -x_C\}, \quad A_{q_{\perp}} = B\{y_q,\ -x_q\}\ ,\quad
A_{\zeta _{\perp}}= B\{y_{\zeta},\ -x_{\zeta}\} \ .
$$
At the same time, the  pseudomomentum operator ${\cal P}_\perp $ can 
be written as
\begin{equation}
{\cal P}_\perp = (i\nabla _{0\perp } - A_{0\perp}) +(i\nabla 
_{r_{C\perp}} - A_{C\perp}). \label{12.1}
\end{equation}
Notice that the operator ${\cal P}_\perp$ does not depend on $q_{\perp}$ 
and  $\zeta_{\perp}$. Fixing the pseudomomentum does not effect the 
last two terms on the r.h.s. of~(\ref{11.2}). The first two terms on 
the r.h.s. of~(\ref{11.2}) coinside with operator $T_{\perp}$ for an 
effective two-particle system, consisting of the nucleus and an 
effective particle with the charge $-3$ and mass $\frac3{2}$. 
The operator ${\cal P}_\perp$ also coinsides with the operator of 
pseudomomentum for this effective two-particle system. We will use 
these observations to rewrite the operator $T_{\perp \nu}$ in a more 
convenient form.

For fixed $\nu = (\nu _1,\ \nu _2)$, let us introduce 
the following~\cite{VZh} coordinates $t_1,\ t_2$
\begin{equation}
  t_1: = x_0 -x_C + \frac3{2} \nu _2 B^{-1},\quad t_1: = y_0 -y_C - 
\frac3{2} \nu _1 B^{-1}\ .  \label{12.2}
\end{equation}
Let $M_R = \frac{3M_0}{M_0-3}$.\ 
The operator $T_{\perp \nu}$ in new variables can be written as
\begin{eqnarray}
T_{\perp \nu}(B) & = & (i\nabla _{q_{\perp}} + A_{q_{\perp}})^2 + 
(i\nabla _{\zeta _{\perp}} + A_{\zeta _{\perp}})^2 - \nonumber \\
& - & \frac2{M_0} \Biggl(\frac{\partial ^2}{\partial t_1^2} + 
\frac{\partial ^2}{\partial t_2^2} \Biggr) +
M_R^{-1}\Biggl\{\Biggl( i \frac{\partial }{\partial t_1}+ 
3Bt_2\Biggr)^2 + \Biggl( i \frac{\partial }{\partial t_2}- 
3Bt_1\Biggr)^2\Biggr\}  \\
& + & \frac{18}{M_0}B^2(t_1^2 + t_2^2)\ . \nonumber
\end{eqnarray}
Notice that the operator $T_{\perp \nu}$ contains the term
\begin{equation}
- \frac2{M_0} \Biggl(\frac{\partial ^2}{\partial t_1^2} + 
\frac{\partial ^2}{\partial t_2^2} \Biggr) +
\frac{18}{M_0}B^2(t_1^2 + t_2^2)
\end{equation}
similar to the Hamiltonian of a harmonic oscilator. Due to this term, 
the energy of the system tends to $\infty$ when the distance betweeen 
the nucleus and the center of masses of the electronic system in a 
direction orthogonal to the magnetic field tends to $\infty$ . 
Theorem 3.1 will be proved in two steps. First, let us consider the 
subsystem $C_1 = (0, \ 1)$, and let $\tilde H_{\nu}^{C_1} (B)$ be the 
operator defined by~(\ref{8.1}) with $C$ replaced  by $C_1$ and 
$V^{C_1}= -3|r_0 -r_1|^{-1}.$
The first step in the proof of Theorem 3.1 is the following
\begin{lemma}
The lowest spectral point $\mu _1$ of the operator $\tilde 
H_{\nu}^{C_1}(B)$ is an infinitely degenerate isolated eigenvalue.
\end{lemma}

{\em Proof.} Notice that the operator $\tilde H_{\nu}^{C_1}(B)$ can be written as
\begin{equation}
\tilde H_{\nu}^{C_1}(B) = h_{\nu}[C_1]\otimes I({\mathbb R}^2) 
+I({\mathbb R}^5)\otimes (i\nabla _{q_{\perp}}+A_{q_{\perp}})^2\ , 
\label{14.1}
\end{equation}
where $I({\mathbb R}^2)$ and $I({\mathbb R}^5)$ are the identity 
operators in ${\cal L}_2({\mathbb R}^2)$ and ${\cal L}_2({\mathbb 
R}^5)$ respectively,

\begin{equation}
h_{\nu}[C_1] = T_{\perp \nu}^{C_1}(B)- \Delta _{oz}[C_1] - 3|r_0 - r_1|^{-1}\ ,
\end{equation}
\begin{eqnarray}
T_{\perp \nu}^{C_1}(B) & = & (i\nabla _{\zeta _{\perp}} + A_{\zeta 
_{\perp}})^2 - \frac2{M_0} \Biggl(\frac{\partial ^2}{\partial t_1^2} 
+ \frac{\partial ^2}{\partial t_2^2} \Biggr) \nonumber \\
& + & M_R^{-1}\Biggl\{\Biggl( i \frac{\partial }{\partial t_1}+ 
3Bt_2\Biggr)^2 + \Biggl( i \frac{\partial }{\partial t_2}- 
3Bt_1\Biggr)^2\Biggr\}  \\
& + & \frac{18}{M_0}B^2(t_1^2 + t_2^2)\ . \nonumber
\end{eqnarray}
Due to~(\ref{14.1}),  it suffices to show that the 
lowest spectral point of the operator $h_{\nu}[C_1]$ belongs to its 
discrete spectrum, in order  to prove Lemma 5.1.

Notice that the operator $T_{\perp \nu}^{C_1}(B)$ has pure point 
spectrum, because the operator $(i\nabla _{\zeta _{\perp}} + A_{\zeta 
_{\perp}})^2 $ has isolated infinitely degenerate eigenvalues and the 
part of $T_{\perp \nu}^{C_1}(B)$, depending on $t_1$ and $t_2$ is an 
operator with pure discrete spectrum. Let $\kappa [C_1]= \inf \sigma 
(T_{\perp \nu}^{C_1}(B))$ and $\varphi _{\perp}(t_1,\ t_2,\ \zeta 
_{\perp})$ be a normalized eigenfunction of $T_{\perp \nu}^{C_1}(B)$, 
corresponding to $\kappa [C_1]$.

\begin{lemma}
\begin{eqnarray}
& i) & \inf \sigma (h_{\nu}[C_1]) < \kappa [C_1] \label{15.1}\\
& ii) &  For\ an\ arbitrary\  \delta >0\
\ one\ can\ find\ a\ finite\ dimentional\ subspace\nonumber \\
& &{\cal M} \subset {\cal L}_2({\mathbb R}^5),\ \
  such\ that\ for\ all\quad \psi \in D(h_{\nu}[C_1]),\quad \psi \perp 
{\cal M}\ \ holds \nonumber\\
& & (h_{\nu}[C_1]\psi ,\ \psi ) \ge (\kappa [C_1] - \delta )\| \psi 
\|^2\ .\label{15.2}
\end{eqnarray}
%\begin{equation}
%(h_{\nu}[C_1]\psi ,\ \psi ) \ge (\kappa [C_1] - \delta )\| \psi 
%\|^2\ .\label{15.2}
%\end{equation}
\end{lemma}

With the remarks above, Lemma 5.2 implies Lemma 5.1.

{\em Proof of Lemma 5.2.} Let $\xi _z = z_0 - z_1$,
\begin{equation}
\psi _0 = \varphi _{\perp} (t_1, t_2, \zeta _{\perp})f(\xi _z)\ , \label{15.3}
\end{equation}
where the function $f(\xi _z)$ will be chosen later. Obviously, 
$\Delta _{0z}[C_1]= \frac{M_0+1}{M_0}\ \frac{\partial ^2}{\partial 
\xi ^2_z}$ and
\begin{equation}
(h_{\nu}[C_1]\psi _0, \ \psi _0) = \kappa [C_1]\|\psi _0\|^2 + 
\frac{M_0+1}{M_0}\ \|\nabla _{\xi _z}f \|^2 - 3\int |r_1 - r_0|^{-1} 
|\varphi _{\perp}|^2 |f|^2 d\Omega \ . \label{16.1}
\end{equation}
The potential $|r_1 - r_0|^{-1}$ in variables $\xi _z$, $t_1$, $t_2$, 
$\zeta _{\perp}$ can be written as
\begin{equation}
|r_1 - r_0|^{-1} = \Biggl[\xi _z^2 + \Biggl(t_1 - \frac{3\nu 
_2}{2B}+\frac2{3}x_{\zeta}\Biggr)^2 + \Biggl(t_2 + \frac{3\nu 
_1}{2B}-\frac2{3}y_{\zeta}\Biggr)^2 \Biggr]^{-\frac1{2}}\ . 
\label{16.2}
\end{equation}
Let
\begin{equation}
{\cal V}(\xi _z) = -3\int|\varphi _{\perp}(t_1,\ t_2,\ \zeta 
_{\perp})|^2\ |r_1 - r_0|^{-1}dt_1dt_2d\zeta _{\perp}\ . \label{16.3}
\end{equation}
For large $|\xi _z|$, the function ${\cal V}(\xi _z)$ decays like $|\xi 
_z|^{-1}$ . Consider the operator
\begin{equation}
-\frac{d^2}{d\xi _z^2} + {\cal V}(\xi _z) \label{16.4}
\end{equation}
on a half-line $\xi _z\ge 1$ with  Dirichlet boundary conditions 
at $\xi _z = 1$. The discrete spectrum of this operator is non-empty, 
and to prove~(\ref{15.1}, it suffices to choose as $f(\xi _z)$ 
in~(\ref{15.3}) an eigenfunction of~(\ref{16.4}) for $\xi _z \ge 1$ 
and $0$ for $\xi _z < 1$.

Let us proceed to the proof of statement ii) of Lemma 5.2. To prove it,  
we will show that for the states supported outside some compact 
region,~(\ref{15.2}) holds. The arguments behind the proof are very 
simple. If $|\xi _z|$ or $|\zeta _{\perp}|$ is large, the potential 
$-3|r_0 -r_1|^{-1}$ is small, and~(\ref{15.2}) holds. If $|t_1|$ or 
$|t_2|$ is large, then the quadratic form of the operator
\begin{equation}
\frac2{M_0}\Biggl(\frac{\partial ^2}{\partial t_1^2} + 
\frac{\partial ^2}{\partial t_2^2}\Biggr) + \frac{18}{M_0}(t_1^2 + 
t_2^2) \label{17.1}
\end{equation}
is large and~(\ref{15.2}) holds also.

For $t \in {\mathbb R}_+^1$ let us define smooth functions $u(t)$ and 
$v(t)$ such that $u^2 + v^2 = 1$ , $u(t)= 1$ for $t\le 1$ and 
$u(t)=0$ for $t\ge 2$. Let for $a>0$ $u_1 = u((t_1^2 + 
t_2^2)^{1/2}\cdot a^{-1})$. Then, for $\psi \in C_0^2({\mathbb R}^5)$,
\begin{eqnarray}
(h_{\nu} [C_1]\psi ,\ \psi ) & \ge & (h_{\nu} [C_1]\psi u_1 ,\ \psi 
u_1) + (h_{\nu} [C_1]\psi v_1 ,\ \psi v_1) \nonumber \\
& - & C\int \{|\nabla u_1|^2 + |\nabla v_1|^2\} |\psi |^2 d\Omega  \ 
. \label{17.2}
\end{eqnarray}
Notice that
\begin{equation}
|\nabla u_1|^2 + |\nabla v_1|^2 \le a^{-2}\max\limits_{t\in [1,2]}\{ 
|u_t^{\prime}|^2 + |v_t^{\prime}|^2\}  \ ,  \label{18.1}
\end{equation}
which implies for
$$
a> \Biggl[\frac{\delta}{2} \max\limits_t \{ u_t^{\prime 2}  + 
v_t^{\prime 2} \}  \Biggr]^{\frac1{2}}
$$
the inequality
\begin{equation}
(h_{\nu} [C_1]\psi ,\ \psi ) \ge L_[\psi u_1] + L_1[\psi v_1] \
\label{18.2}
\end{equation}
with
\begin{equation}
L_1[\varphi ]: = (h_{\nu} [C_1]\varphi ,\ \varphi ) - 
\frac{\delta}{2}\|\varphi \|^2 \ . \label{18.3}
\end{equation}
First, we would like to estimate the term $L_1[\psi v_1]$.

Let
\begin{equation}
h^1_{\nu} [C_1] = h_{\nu}[C_1] + M_0^{-1}\Biggl(\frac{\partial 
^2}{\partial t_1^2} +  \frac{\partial ^2}{\partial t_2^2}\Biggr) - 
M_0^{-1}B^2(t_1^2 + t_2^2)\ . \label{18.4}
\end{equation}
The operator $h^1_{\nu} [C_1]$ is semibounded from below, and the 
quadratic form of the operator
\begin{equation}
- M_0^{-1}\Biggl(\frac{\partial ^2}{\partial t_1^2} +  \frac{\partial 
^2}{\partial t_2^2}\Biggr) + M_0^{-1}B^2(t_1^2 + t_2^2)  \label{18.5}
\end{equation}
tends to $+\infty$ on functions supported outside the ball $\{(t_1,\ 
t_2)| t_1^2 + t_2^2 \le a^2 \}$ as $a \to \infty$. Together 
with~(\ref{18.5}), this implies
\begin{equation}
L_1[\psi v_1] > \kappa \| \psi v_1 \|^2  \label{19.1}
\end{equation}
for large $a>0$.

Let us turn to the estimate of $L_1[\psi u_1]$. For this term, we 
shall introduce an additional partition of unity of the configuration 
space, this time in variables $\zeta _{\perp}$ and $\xi _z$. Let $u_2 
= u ((|\zeta _{\perp}|^2 + |\xi _z|^2)^{\frac1{2}}\cdot a^{-1}) $, 
$v_2 = (1-u_2^2)^{\frac1{2}}$. Similar to~(\ref{17.2}) 
and~(\ref{18.2}) we arrive at
\begin{equation}
L_1[\psi u_1] \ge L_2[\psi u_1u_2] + L_2[\psi u_1v_2]\ , \label{19.2}
\end{equation}
where
\begin{equation}
L_2[\varphi ]: = L_1[\varphi ] - \frac{\delta}{4}\| \varphi \|^2 \ . 
\label{19.3}
\end{equation}
On the support of $\psi u_1v_2$ for large $a>0 $ holds $3|r_1 - 
r_0|^{-1}< \frac{\delta}{4}$ ,  which implies
\begin{equation}
L_2[\psi u_1v_2] \ge (\kappa - \delta )\|\psi u_1v_2 \|^2 \ .  \label{19.4}
\end{equation}
On the other hand, the function $\psi u_1u_2$ is supported in a compact 
region. The operator $h_{\nu}[C_1]$, defined in this region with  
Dirichlet boundary conditions, has purely discrete spectrum, which 
accumulates at $+\infty$. To complete the proof of Lemma 5.2, it 
suffices now to take as the subspace $\cal M $ a linear span of the 
$N$ first eigenfunctions of this operator multiplied by $u_1\cdot 
u_2$ for sufficiently large $N$. $\Box$

Now we are prepared to start directly with the proof of Theorem 3.1. 
Obviously, it suffices to prove the following statement very similar 
to Lemma 5.2:
\begin{lemma}
\begin{eqnarray}
& i) & \inf \sigma (\tilde H_{\nu}^C(B)) < \mu _1  \label{20.1} \\
& ii) & For\ an\ arbitrary\ \delta >0,\ one\ can\ find\ a\ finite\ 
dimentional\ \nonumber \\
& & subspace\ {\cal M}_1 \subset {\cal L}_2({\mathbb R}^8),\
such\ that\ for\ all\ \psi \in D(\tilde H_{\nu}^C(B))\ ,\ \psi \perp 
{\cal M}_1\  holds \nonumber\\
& & (\tilde H_{\nu}^C(B)\psi ,\ \psi ) \ge (\mu _1 - \delta )\| \psi 
\|^2\ .\label{20.2}
\end{eqnarray}
%\begin{equation}
%(\tilde H_{\nu}^C(B)\psi ,\ \psi ) \ge (\mu _1 - \delta )\| \psi 
%|^2\ .\label{20.2}
%\end{equation}
\end{lemma}

{\em Proof of Lemma 5.3.}  Let $\varphi (t_1,\ t_2,\ \zeta _{\perp}, \ 
q_{\perp}, \ \xi _z)$ be an eigenfunction of $\tilde H_{\nu}^C(B)$, 
corresponding to the eigenvalue $\mu _1$, $\eta _z = z_2 - \frac{z_1 
+ M_0z_0}{M_0+1}$ and
\begin{equation}
\psi _0(t_1,\ t_2,\ \zeta _{\perp}, \ q_{\perp}, \ \xi _z, \ \eta _z) 
= \varphi _0 (t_1,\ t_2,\ \zeta _{\perp}, \ q_{\perp}, \ \xi _z) 
f_0(\eta _z)u_3\ , \label{21.2}
\end{equation}
where $u_3 = u(|r_0 - r_2|\cdot |\eta _z|^{-1})$, the function $u(t)$ 
is the same as in the proof of Lemma 5.2 and function $f_0(\eta _z)$ 
will be chosen later.

To prove statement $i)$, it suffices to show that for a suitable 
choice of $f(\eta _z)$ the inequality
$$
(\tilde H_{\nu}^C(B)\psi ,\ \psi )< \mu _1\| \psi \|^2
$$
holds.

Notice that in variables $(\xi _z, \ \eta _z)$, the operator $\Delta 
_{0z}[C]$ can be written as
\begin{equation}
\Delta _{0z}[C] = \frac{M_0 + 1}{M_0}\ \frac{\partial ^2}{\partial 
\xi ^2_z} + \frac{M_0 + 2}{M_0 + 1}\ \frac{\partial ^2}{\partial \eta 
^2_z} \ .  \label{21.2a}
\end{equation}
On the support of the function $u(8|r_0 - r_1|\ |\eta _z|^{-1})$, the 
distance between the first electron and the nucleus is at least three 
times smaller than between the subsystem $C_1$ and the second 
electron.

Let us estimate the quadratic form
\begin{eqnarray}
(\tilde H_{\nu}^C(B)\psi _0,\ \psi _0) & = & (\tilde 
H_{\nu}^{C_1}(B)\psi _0,\ \psi _0) - \frac{M_0 + 2}{M_0 + 1}(\Delta 
_{\eta _z}\psi _0,\ \psi _0 ) \nonumber \\
& - & 3(|r_0 -r_1|^{-1}\psi _0,\ \psi _0 ) + (|r_1 -r_2|^{-1}\psi 
_0,\ \psi _0 ) . \label{22.1}
\end{eqnarray}
According to~(\ref{16.2}) $\ \ |r_0 -r_1|$ does not depend on $\eta _z$ and
\begin{equation}
\Biggl| \frac{\partial }{\partial \eta _z}  u(8|r_0 - r_1|\cdot |\eta 
_z|^{-1})\Biggr| \le \max\limits_{t\in [0,\ 2]}|u_t^{\prime}|\cdot 
2|\eta _z|^{-1} = C_1\cdot |\eta _z|^{-1}\ .  \label{22.2}
\end{equation}
Assume $f(\eta _z) = 0$ for $|\eta _z| \le 1$. Then for $\varepsilon >0$
\begin{equation}
-\Biggl( \frac{\partial ^2}{\partial \eta ^2_z} \varphi _1 f u_3,\ 
\varphi _1 f u_3\Biggr) \le (1 + \varepsilon )\| u_3\cdot 
f^{\prime}(\eta _z)\|^2 + (1 + \varepsilon ^{-1}) C_1^2 \|f\cdot 
|\eta _z|^{-1}  \|^2 \ .   \label{22.3}
\end{equation}
On the support of $\psi _0$ the inequality
\begin{equation}
-3|r_0 -r_1|^{-1} + |r_1 -r_2|^{-1} \le -\frac3{2} |r_2 -r_0|^{-1} 
\label{22.4}
\end{equation}
holds and the last two terms on the r.h.s. of~(\ref{22.1}) can be 
estimated from above as
\begin{equation}
-\frac3{2} (|r_2 -r_0|^{-1}\psi _0,\ \psi _0)\ .  \label{22.5}
\end{equation}
Notice that
\begin{eqnarray}
|r_0 -r_2| & = & \Biggl[\Biggl(-\eta _z + \frac1{M_0 +1}\ \xi _z 
\Biggr)^2 + \Biggl(\frac1{2}x_{q_{\perp}} - \frac1{3}x_{\zeta 
_{\perp}} + t_1 - \frac{3\nu _2}{2B}\Biggr)^2 \nonumber \\
& + & \Biggl(\frac1{2}y_{q_{\perp}} - \frac1{3}y_{\zeta _{\perp}} + 
t_2 + \frac{3\nu _1}{2B}\Biggr)^2 \ \Biggr]^{\frac1{2}} \ . 
\label{22.6}
\end{eqnarray}
Let
\begin{equation}
{\cal V}_1(\eta _z) = -\frac3{2}\int |r_2 -r_0|^{-1}|\varphi _0 
(t_1,\ t_2,\ \xi _z, \ \zeta _{\perp}, \ q_{\perp})|^2\cdot |u(8|r_0 
- r_1|\cdot |\eta _z|^{-1} )|^2 \ dt_1 dt_2 d\xi _z d\zeta _{\perp} 
dq_{\perp} \ .  \label{23.1}
\end{equation}
For large $|\eta _z|$
$$
{\cal V}_1(\eta _z) \le -\frac5{4}|\eta _z|^{-1}\ .
$$
Combining~(\ref{22.3}) and~(\ref{23.1}) we arrive at
\begin{eqnarray}
(\tilde H_{\nu}^C(B)\psi _0,\ \psi _0) & = & (\tilde 
H_{\nu}^{C_1}(B)\psi _0,\ \psi _0) + \frac{M_0 + 2}{M_0 + 1}(1 + 
\varepsilon )\|f^{\prime}(\eta _z) \| \nonumber \\
& + & \int {\cal V}_2(\eta _z) |f(\eta _z)|^2 \ d\eta _z\ ,  \label{23.2}
\end{eqnarray}
where ${\cal V}_2(\eta _z) \le - |\eta _z|^{-1}$ for large $|\eta 
_z|$. Let us turn to the estimate of the upper bound of $(\tilde 
H_{\nu}^{C_1}(B)\psi _0,\ \psi _0)$. For an eigenfunction $g$ of the 
operator $h[C_1]$ , corresponding to its lowest eigenvalue, we have~\cite{Zh}
\begin{equation}
|r_0 - r_1|g \in {\cal L}_2({\mathbb R}^5)  \ , \label{23.3}
\end{equation}
which implies also
\begin{equation}
|r_0 - r_1|{\varphi}_0 \in {\cal L}_2({\mathbb R}^7)  \ . \label{23.4}
\end{equation}
Using~(\ref{23.4}) and semiboundedness from below of the operator 
$\tilde H_{\nu}^C(B)$ in a standard way, (see~\cite{VZh1}, 
~\cite{VZh2}, ~\cite{Zh}) we arrive at
\begin{eqnarray}
(\tilde H_{\nu}^C(B)\psi _0,\ \psi _0) & \le & (\tilde 
H_{\nu}^{C_1}(B)\varphi _0f,\ \varphi _0f) + C\int |\eta 
_z|^{-2}|f(\eta _z) |^2\ d\eta _z  \nonumber \\
& = & \mu _1\|f(\eta _z)\|^2 + C\int |f(\eta _z)|^2 \ |\eta _z|^{-2} 
\ d\eta _z\ .   \label{24.1}
\end{eqnarray}
To complete the proof of the statement $i)$ of Lemma 5.3, it suffices 
now to combine the estimates~(\ref{24.1}) and~(\ref{23.2}), and to 
mention that the discrete spectrum of the operator
\begin{equation}
-\frac{M_0 + 2}{M_0 + 1}(1 + \varepsilon )\frac{\partial ^2}{\partial 
\eta ^2_z} + {\cal V}_2(\eta _z) + C|\eta _z|^{-2}   \label{24.2}
\end{equation}
defined in ${\cal L}_2([1,+\infty ))$ with  Dirichlet boundary 
condition at $\eta _z = 1 $ is nonempty. Thus we can take $f(\eta 
_z)$ equal to an eigenfunction of~(\ref{24.2}) on$[1,+\infty )$, and 
$0$ otherwise.

Our next goal is to prove statement $ii)$ of Lemma 5.3. First, let us 
discuss heuristically the arguments behind the proof. To prove statement 
$ii)$ we shall show that outside some compact region in variables $(t_1,\ 
t_2,\ \zeta _{\perp} , \ q_{\perp}, \ \xi _z, \ \eta _z)$,  the 
inequality~(\ref{20.2}) holds.
Let
$$
\Omega _1(a_1) = \{ (t_1,\ t_2,\ \zeta _{\perp} , \ q_{\perp}, \ \xi 
_z, \ \eta _z)\ | \quad t_1^2 + t_2^2 \ge a_1 \} \ ,
$$
$$
\Omega _2(a_1,\ a_2) = \{ (t_1,\ t_2,\ \zeta _{\perp} , \ q_{\perp}, 
\ \xi _z, \ \eta _z)\ | \quad t_1^2 + t_2^2 < a_1,\ |\zeta 
_{\perp}|^2 + |q_{\perp}|^2 \ge a_2 \}
$$
and
$$
\Omega _3(a_1,\ a_2,\ a_3) = \{ (t_1,\ t_2,\ \zeta _{\perp} , \ 
q_{\perp}, \ \xi _z, \ \eta _z)\ | \quad t_1^2 + t_2^2 < a_1,\ |\zeta 
_{\perp}|^2 + |q_{\perp}|^2 < a_2 ,\ |\xi _z|^2 + |\eta _z|^2\ge 
a_3\}\ .
$$
Notice that the complement of the region $\Omega _1\bigcup \Omega 
_2\bigcup \Omega _3$ is a compact region. Assume that the function $\psi$ 
is supported in $\Omega _1$. Then, as we have seen already in the 
proof of Lemma 5.2, the quadratic form of the operator $T_{\perp \nu} $ 
is large for large $a_1$ and the inequality~(\ref{20.2}) holds. 
Assume now that $\psi $ is supported in $\Omega _2(a_1,\ a_2)$ and 
$a_2\gg a_1$. In this case, at least two of three electrons are very 
far from the nucleus, which implies that at least one of the 
potentials $-3|r_0 - r_1|^{-1}$ or $-3|r_0 - r_2|^{-1}$ is less than 
$\frac{\delta}2$ and the inequality~(\ref{20.2}) holds also.

The same argument works also if $\psi$  is supported in $\Omega _3$. 
The inequality $|\xi _z|^2 + |\eta _z|^2\ge a_3$ ensures that one of 
the potentials $-3|r_0 - r_1|^{-1}$ or $-3|r_0 - r_2|^{-1}$ is small.

To implement the idea described in the previous subsection, let us 
define the following localization functions:
\begin{eqnarray}
& i) & u_4(t_1,\ t_2) = u((t_1^2 + t_2^2)^{\frac1{2}}a_1^{-1})\ , 
\label{26.1} \\
&  & v_4 = (1-u_4^2)^{\frac1{2}}\ ;   \label{26.2}\\
& ii) & {\cal U}_1 = u_4 \cdot v((|\zeta _{\perp}|^2) + |q_{\perp}|^2 
)^{\frac{1}2}a_2^{-1})\cdot u\left( \frac{\sqrt{2}}2 |r_0 -r_1| \ 
|r_0-r_2|^{-1} \right)\ , \label{26.3} \\
& & {\cal U}_2 = u_4 \cdot v((|\zeta _{\perp}|^2) + |q_{\perp}|^2 
)^{\frac{1}2}a_2^{-1})\cdot v\left( \frac{\sqrt{2}}2 |r_0 -r_1| \ 
|r_0-r_2|^{-1} \right)\ , \label{26.4} \\
& & {\cal U}_0 = u_4 \cdot u((|\zeta _{\perp}|^2) + |q_{\perp}|^2 
)^{\frac{1}2}a_2^{-1})\ ; \label{26.5} \\
& iii) & {\cal U}_3 = {\cal U}_0 \cdot v((|\zeta _z|^2) + |\eta _z|^2 
)^{\frac{1}2}a_3^{-1})\cdot u\left( \frac{\sqrt{2}}2 |r_0 -r_1| \ 
|r_0-r_2|^{-1} \right)\ ; \label{27.1} \\
& & {\cal U}_4 = {\cal U}_0 \cdot v((|\zeta _z|^2) + |\eta _z|^2 
)^{\frac{1}2}a_3^{-1})\cdot v\left( \frac{\sqrt{2}}2 |r_0 -r_1| \ 
|r_0-r_2|^{-1} \right)\ ; \label{27.2} \\
& & {\cal U}_5 = {\cal U}_0 \cdot u((|\zeta _z|^2) + |\eta _z|^2 
)^{\frac{1}2}a_3^{-1})\ . \label{27.3}
\end{eqnarray}
Obviously,
\begin{equation}
v_4^2 + \sum\limits_{i=1}^5 {\cal U}_i^2 = 1  \label{27.4}
\end{equation}
and the function ${\cal U}_5$ is compactly supported.

Standard localization error estimates show that for large $a_1$, 
$a_2$, $a_3 > 0$ and an arbitrary function $\psi \in {\cal D}(\tilde 
H_{\nu}^C (B))$ holds
\begin{equation}
(\tilde H_{\nu}^C (B)\psi ,\ \psi ) \ge (\tilde H_{\nu}^C (B)\psi 
v_4,\ \psi v_4) + \sum\limits_{i=1}^5 (\tilde H_{\nu}^C (B)\psi {\cal 
U}_i,\ \psi  {\cal U}_i) - \frac{\delta}2 \| \psi \|^2   \ . 
\label{27.5}
\end{equation}
The estimate for the functional $(\tilde H_{\nu}^C (B)\psi v_4,\ \psi 
v_4)$ is very similar to one given in the proof of Lemma 5.2.

Analogously to (\ref{18.4}) we define operator
\begin{equation}
\bar H_{\nu}^C (B) = \tilde H_{\nu}^C (B) + M_0^{-1} 
\left(\frac{\partial ^2}{\partial t_1^2} + \frac{\partial 
^2}{\partial t_2^2} \right) - M_0^{-1}B^2(t_1^2 + t_2^2) \ , 
\label{28.1}
\end{equation}
which is semibounded from below, and use the fact, that the infimum 
of the quadratic form of the operator~(\ref{18.5}) on functions 
supported outside the region $\{(t_1,\ t_2)\ |\ t_1^2 + t_2^2 \ge 
a_1^2 \}$ tends to~$+\infty$ as $a_1 \to +\infty$. This implies
\begin{equation}
(\tilde H_{\nu}^C (B)\psi v_4,\ \psi v_4)\ge \mu _1\|\psi v_4 \|^2 
\label{28.1a}
\end{equation}
for large $a_1$.

The functions $\psi{\cal U}_1$ and $\psi{\cal U}_2$ can be obtained 
one from another by permutation of electrons. It suffices to estimate 
the quadratic form of $\tilde H_{\nu}^C (B)$ on one of them. We shall 
do it for the function $\psi{\cal U}_1$.

On the support of $\psi{\cal U}_1$ the following inequalities hold:
\begin{eqnarray}
& i) & t_1^2 + t_2^2 \le 4a_1^2\ ,   \label{29.1} \\
& ii) & |\zeta _{\perp}|^2 +|q_{\perp}|^2 \ge a_2^2\ ,   \label{29.2} \\
& iii) & |r_0 - r_1|\le \sqrt{2}|r_0 -r_2|\ .   \label{29.3}
\end{eqnarray}
According to (\ref{12.2}) on the support of $\psi{\cal U}_1$
\begin{eqnarray}
|r_{0\perp} -r_{C\perp}| &  =  & \left[ \left(t_1 - \frac{3\nu 
_2}{2B}\right)^2 + \left(t_2 - \frac{3\nu 
_1}{2B}\right)^2\right]^{\frac1{2}} \nonumber \\
& \le & [2(t_1^2 + t_2^2) + \frac9{2B^2}(\nu _1^2 + \nu 
_2^2)]^{\frac1{2}}\nonumber \\
& \le & 2\sqrt{2} a_1 + \frac3{\sqrt{2}B}(\nu _1^2 + \nu 
_2^2)^{\frac1{2}}\ . \label{29.4}
\end{eqnarray}
On the other hand,
\begin{equation}
|r_{1\perp} -r_{C\perp}| + |r_{2\perp} -r_{C\perp}| \ge |\zeta 
_{\perp}| + \frac1{2}\Bigl||q_{\perp}| -|\zeta _{\perp}|\Bigr| \ge 
\frac1{2}\Bigl(|\zeta _{\perp}| + |q_{\perp}| \Bigr) \ge 
\frac1{2}a_2\ , \label{29.5}
\end{equation}
which together with (\ref{29.4}) implies
\begin{equation}
|r_{1\perp} -r_{0\perp}| + |r_{2\perp} -r_{0\perp}| \ge  \frac1{2}a_2 
-4\sqrt{2}a_1 -\frac{3\sqrt{2}}{B}(\nu _1^2 + \nu _2^2)^{\frac1{2}}\ 
. \label{29.6}
\end{equation}
Now we assume that the number $a_2$ is chosen so large that
$$
\frac1{4}a_2 \ge 4\sqrt{2}a_1 + \frac{3\sqrt{2}}{B}(\nu _1^2 + \nu 
_2^2)^{\frac1{2}}\ .
$$
Then
\begin{equation}
|r_1 - r_0| + |r_2 - r_0| \ge  |r_{1\perp} -r_{0\perp}| + |r_{2\perp} 
-r_{0\perp}| \ge  \frac1{4}a_2 \ , \label{29.7}
\end{equation}
and due to (\ref{29.3})
\begin{equation}
|r_0 - r_2| \ge \frac1{\sqrt{2}} |r_0 - r_1| \ge \frac{1}{1+\sqrt{2}} 
\cdot \frac1{4}a_2 \ . \label{30.1}
\end{equation}
The last inequality shows that $a_2$ can be chosen so large that 
$3|r_0 - r_2|^{-1} < \frac1{2}\delta$ and
\begin{equation}
(\tilde H_{\nu}^C (B)\psi {\cal U}_1,\ \psi  {\cal U}_1) \ge \inf 
\sigma (\tilde H_{\nu}^{C_1} (B)) \cdot \| \psi {\cal U}_1\|^2 - 
\frac{\delta}2\| \psi {\cal U}_1\|^2 = (\mu _1  - \frac{\delta}2)\| 
\psi {\cal U}_1\|^2  \label{30.2}
\end{equation}
Let us now prove  that
\begin{equation}
(\tilde H_{\nu}^C (B)\varphi,\ \varphi) \ge (\mu _1  - 
\frac{\delta}2)\| \varphi\|^2  \label{30.3}
\end{equation}
for $\varphi = \psi {\cal U}_3$ and $\varphi = \psi {\cal U}_4$ . 
Again, $ \psi {\cal U}_3$ can be obtain from $\psi {\cal U}_4$ by 
permutation of the electrons, and it suffices to prove the 
inequality~(\ref{30.3}) for one of these functions.

On the support of $\psi {\cal U}_3$, the inequalities
\begin{eqnarray}
& i) &  |\xi _z|^2 + |\eta _z|^2 \ge a_3^2  \qquad {\rm and}\label{31.1}\\
& ii) &  |r_0 - r_2|\ge \frac1{\sqrt{2}}|r_0 -r_1|\label{31.2}
\end{eqnarray}
hold. Recall that
$$
\xi _z = z_0 -z_1 \quad {\rm and} \quad \eta _z = z_2 - \frac{z_1 + 
M_0z_0}{M_0+1} = z_2 - z_0 + \frac{z_0 - z_1}{M_0+1}\ .
$$
Thus we have
\begin{equation}
|z_0 - z_1|+ |z_0 - z_2| \ge |\xi _z| + |\eta _z| - \frac{|\xi 
_z|}{M_0 + 1}\ge \frac{M_0}{M_0+1}|\xi _z| + |\eta _z|\ge 
\frac{M_0}{M_0+1}\ a_3\ , \label{31.3}
\end{equation}
which implies
\begin{eqnarray}
|r_0 - r_2| & \ge & \frac1{1+\sqrt{2}}\{|r_0 - r_1| + |r_0 - r_2|\} 
\nonumber \\
& \ge &  \frac1{1+\sqrt{2}}\{|z_0 - z_1| + |z_0 - z_2|\} \ge 
\frac{M_0}{(1+\sqrt{2}){(M_0 +1)}}\ a_3\ . \label{31.4}
\end{eqnarray}
For large $a_3$ similar to~(\ref{30.2}) we arrive at
\begin{equation}
(\tilde H_{\nu}^C (B)\psi {\cal U}_3,\ \psi  {\cal U}_3) \ge (\mu _1 
- \frac{\delta}2) \|\psi {\cal U}_3 \|^2 \ . \label{31.5}
\end{equation}
To complete the proof of Lemma 5.3 and Theorem 3.1, it suffices to 
notice that the function $\psi {\cal U}_5$ is supported in a compact 
region
$$
\Omega _0 = \{(t_1,\ t_2,\ \zeta _{\perp}, \ q_{\perp },\ \xi _z, \ 
\eta _z)\ | \ t_1^2 + t_2^2 \le 4a_1^2\ , \ |\zeta _{\perp}|^2 + 
|q_{\perp }|^2 \le 4a_2^2\ . |\xi _z|^2 + |\eta _z|^2 \le 4a_3^2 \}\ .
$$
The operator $\tilde H_{\nu}^C (B)$ defined in this region with  
Dirichlet boundary condition, has purely discrete spectrum. Let 
$\varphi _1,\ \ldots ,\varphi _N $ be the $N$-first eigenfunctions of 
this operator. For the subspace ${\cal M}_1$ in Lemma 5.3, one can chose 
the linear span of the functions $\varphi _1{\cal U}_5$,  $\varphi 
_2{\cal U}_5$, $\ldots \ \varphi _N{\cal U}_5$ with sufficiently 
large $N$. $\Box$

\begin{acknowledgements}
The author acknowledges the financial support from ESI Vienna and the 
European project HPRN-CT-2002-00277.
\end{acknowledgements}



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\end{document}

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