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Multiparticle Schr\"odinger operators, homogeneous magnetic field,
pseudomomentum.
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\begin{document}
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\DOIsuffix{theDOIsuffix}
\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.}
%%
\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
{sC_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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