\section{Weighted removal and spectral bounds}\label{sec:03-removal}

This section collects the tools shared by the later sections. We first
record two standard spectral results. We then state a lower bound for the
Coulomb form on regions where the electrons are separated radially
(Lemma~\ref{lem:regional-bound}). Finally we prove the criterion for
nonbinding that Sections~\ref{sec:04-helium-dianion}
and~\ref{sec:07-lithium-dianion} apply to the two dianions
(Proposition~\ref{prop:quadratic-removal}).

The first spectral result is the HVZ theorem. In the present normalization,
its fermionic form is stated in
\cite{nam2012}.

\begin{theorem}[HVZ]\label{thm:hvz}
For the operators in Set-up~\ref{setup:atoms}, every integer $N\geq2$ and
every $Z>0$ satisfy
\[
 \sigma_{\mathrm{ess}}(\AtomH{N}{Z})=[\AtomE{N-1}{Z},\infty).
\]
\end{theorem}

The second spectral result is the spectrum of the hydrogenic operator. The
formula below uses the Hartree kinetic coefficient $1/2$.

\begin{theorem}[Hydrogenic spectrum]\label{thm:hydrogen}
For $Q>0$, let $h_Q=-\Delta/2-Q/|x|$ be the scalar Coulomb operator on
$L^2(\mathbb R^3)$. Its negative eigenvalues are
$-Q^2/(2n^2)$, $n\in\mathbb N$, with spatial multiplicity $n^2$, and its
remaining spectrum is $[0,\infty)$. If $P_m$ projects onto the shells
$1\leq n\leq m$, where $m\geq1$ is an integer, then
\[
 h_Q\big|_{(1-P_m)L^2}\geq-\frac{Q^2}{2(m+1)^2}.
\]
\end{theorem}

This is \cite[Theorem~10.9]{teschl2009}, with its operator
$-\Delta-\gamma/|x|$ taken at $\gamma=2Q$ and divided by two.
After including spin, the first $m$ hydrogenic shells have dimension
$2\sum_{n=1}^m n^2$.

Both dianion proofs need lower bounds for the Coulomb form on regions
where some electrons are confined to a ball and the others are kept
outside a larger ball. We call the confined electrons core electrons and
the others outer electrons. The following lemma collects these bounds. It
uses no symmetry of the test function.

\begin{lemma}[Regional lower bound]\label{lem:regional-bound}
Let $2\le N\le9$, $Z>0$, and let $\mathfrak h$ denote the Coulomb
integral of Set-up~\ref{setup:atoms} with $N$ electrons and charge $Z$
on the full space $H^1((\mathbb R^3)^N;(\mathbb C^2)^{\otimes N})$,
without symmetry restriction. For $-1\le z<1$ put
$c(z)=\min\{1,[2(1-z)]^{-1/2}\}$ and $c(1)=1$, and let
$D_0,\ldots,D_N$ be rational numbers with $D_0=D_1=0$ and
\[
 D_h\le\binom h2\sqrt{\frac{h-1}{2h}}\qquad(2\le h\le N);
\]
the values used in this paper are
\[
 (D_0,D_1,D_2,D_3,D_4,D_5)
 =\left(0,0,\frac12,\frac{433}{250},\frac{1837}{500},\frac{1581}{250}\right),
\]
whose squares for $h=3,4,5$ are below the exact values $3,27/2,40$.
Fix $R,b,\lambda>0$, $\sigma\ge1$, an integer $0\le j\le N-1$, and set
\[
 \kappa=\lambda\sigma^2,\qquad
 C=\frac{\lambda^2}{2}+\frac{\lambda(\sigma-1)^2}{R},\qquad
 A_j=Z-\kappa-\frac{jR}{R+b},
\]
\[
 M_j(R,b,\lambda,\sigma)
 =\frac{jZ^2}{2}+(N-j)C+\frac1R\max_{0\le h\le N-j}\bigl(A_jh-D_h\bigr).
\]
If a spinor $v\in H^1$ is supported where $j$ specified electrons have
radii below $b$ and the other $N-j$ have radii above $R$, then
\[
 \mathfrak h[v]\ge-M_j(R,b,\lambda,\sigma)\,\|v\|^2 .
\]
For $j=0$ the parameter $b$ is immaterial.
\end{lemma}

\emph{Idea of the proof.}
The repulsion among the outer electrons is bounded below in terms of their
directions. Each outer electron then satisfies a one-dimensional radial
inequality, and each core electron is bounded by its hydrogenic ground
energy. The full proof is in
Appendix~\ref{sec:13-deferred-proofs-regional}.

The increments $D_{h+1}-D_h$ of the listed values are
$0,\frac12,\frac{308}{250},\frac{971}{500},\frac{53}{20}$. Hence the
maximum defining $M_j$ is attained at $h=2$ whenever
$\frac12\le A_j\le\frac{308}{250}$, which is the case for every parameter
choice below.

We now state the criterion for nonbinding. Its proof follows Lieb's
argument for the bound $N<2Z+1$~\cite{lieb1984}. Lieb tests the eigenvalue
equation against the eigenfunction multiplied by the radius of one
electron, and bounds the energy of the other $N-1$ electrons from below by
the threshold. We use the quadratic weight $r_i^2+\beta$ in place of the
linear one. We also add correction terms $B_j$. They use the hypothesis
that every state with at most $j-1$ electrons within the radius $b_j$ has
energy at least $d_j$ above $U$. We call $d_j$ an exterior gap.

\begin{proposition}\label{prop:quadratic-removal}
Let $N\geq2$, $Z>0$, $\beta>0$, $U\geq\AtomE{N-1}{Z}$, and let
$J\subset\{1,\ldots,N\}$ be finite. For $j\in J$ let $b_j,d_j>0$ and
$\lambda_j\geq0$. Assume
\[
 \mathfrak q_{N,Z}[v]-U\|v\|^2\geq d_j\|v\|^2
\]
in the quadratic-form sense for every fermionic form-domain vector $v$
vanishing almost everywhere on $\{q_j\leq b_j\}$.
For $s\geq0$ define
\[
 B_j(s)=(s-b_j)_+^2-\frac{\min\{s/b_j,1\}}{2d_j}.
\]
For compactly supported fermionic $u\in H^1$ define
\begin{align*}
 M_{\beta,U}[u]={}&\frac12\sum_i\int(r_i^2+\beta)|\nabla_i u|^2\\
 &+\int\left[\sum_{i<j}\frac{r_i^2+r_j^2+\beta}{d_{ij}}
 -Z\sum_i\left(r_i+\frac\beta{r_i}\right)-\frac{3N}{2}-\beta U\right]|u|^2.
\end{align*}
If some $\varepsilon>0$ satisfies
\[
 M_{\beta,U}[u]+\sum_{j\in J}\lambda_j\int B_j(q_j)|u|^2
 \geq\varepsilon\|u\|^2
\]
for every such $u$, then $\AtomE{N}{Z}=\AtomE{N-1}{Z}$ and
$\AtomH{N}{Z}$ has no nonzero $L^2$ eigenfunction at any energy at or below
$\AtomE{N-1}{Z}$.
\end{proposition}

\begin{proof}
Put $T=\AtomE{N-1}{Z}$ and suppose that
$\AtomH{N}{Z}\Psi=E\Psi$, with $E\leq T\leq U$ and $\|\Psi\|=1$.
Write $X=(x_1,\ldots,x_N)$ and let $\nabla$ denote the full gradient in $X$.

\emph{Ordered-radius moments.}
Fix $j\in J$. Outside the null set of tied or zero radii, the function
$q_j$ agrees locally with one of the $r_i$. Hence $q_j$ is symmetric and
Lipschitz, with $|\nabla q_j|=1$ almost everywhere. For $L>0$ set
$g_L=\min\{(q_j-b_j)_+,L\}$. Both $g_L\Psi$ and $g_L^2\Psi$ belong to
the fermionic form domain. Testing the eigenvalue equation with
$g_L^2\Psi$ and expanding the gradient gives
\[
 \mathfrak q_{N,Z}[g_L\Psi]-E\|g_L\Psi\|^2
 =\frac12\int|\nabla g_L|^2|\Psi|^2.
\]
Since $g_L\Psi$ vanishes on $\{q_j\leq b_j\}$ and $E\leq U$, the assumption
of the proposition bounds the left side below by $d_j\|g_L\Psi\|^2$. On the
right side,
$|\nabla g_L|^2=\mathbf1_{\{b_j<q_j<b_j+L\}}\leq\min\{q_j/b_j,1\}$.
Monotone convergence as $L\to\infty$ therefore gives
\begin{equation}\label{eq:removal-moments}
 \int(q_j-b_j)_+^2|\Psi|^2
 \leq\frac1{2d_j}\int\min\{q_j/b_j,1\}|\Psi|^2.
\end{equation}
In particular, $\int|B_j(q_j)||\Psi|^2<\infty$ and
$\int B_j(q_j)|\Psi|^2\leq0$.

\emph{The weighted identity with a cutoff.}
Choose a real smooth function $\chi:[0,\infty)\to[0,1]$ equal to one
on $[0,1]$ and zero on $[2,\infty)$, and put $C_\chi=\|\chi'\|_\infty$.
For $R\geq1$ set
\[
 \rho=|X|,\qquad \chi_R(X)=\chi(\rho/R),\qquad
 u_R=\chi_R\Psi,\qquad A_R=\{R\leq\rho\leq2R\}.
\]
Then $u_R$ is a compactly supported fermionic $H^1$ vector and
$\|u_R\|\to1$. The gradient $\nabla\chi_R$ is supported in $A_R$ and
satisfies $|\nabla\chi_R|\leq C_\chi/R$.
All multipliers used to test the equation for $\Psi$ below are bounded
with bounded first derivatives. The weak equation extends to
nonfermionic tests by antisymmetrization and permutation invariance.

Test that equation with $r_i^2\chi_R^2\Psi$. Expanding the gradients,
integrating by parts with $\Delta_i r_i^2=6$, and bounding the form of
the remaining $N-1$ electrons below by $T$ gives
\begin{align*}
 &\frac12\int r_i^2|\nabla_i u_R|^2
 +\int\left(-Zr_i+\sum_{k\ne i}\frac{r_i^2}{d_{ik}}-\frac32\right)|u_R|^2\\
 &\qquad\leq(E-T)\int r_i^2|u_R|^2
 +\frac12\int\left(r_i^2|\nabla\chi_R|^2
             +\chi_R\nabla(r_i^2)\cdot\nabla\chi_R\right)|\Psi|^2.
\end{align*}
The bound by $T$ applies to each slice in the other labels, where
$r_i u_R$ is antisymmetric. All terms are finite on the compact support.
The integration by parts is justified by $H^1$ approximation and the
Coulomb form bounds. The unweighted test $\chi_R^2\Psi$ gives
\[
 \mathfrak q_{N,Z}[u_R]-E\|u_R\|^2
 =\frac12\int|\nabla\chi_R|^2|\Psi|^2.
\]
Sum the weighted inequalities and add
$\beta(\mathfrak q_{N,Z}[u_R]-U\|u_R\|^2)$. Since $E\leq T\leq U$
and $\sum_i\nabla(r_i^2)=2X$, this yields
\begin{equation}\label{eq:removal-quadratic-boundary}
\begin{split}
 M_{\beta,U}[u_R]
 &\leq\frac12\int\left((\rho^2+\beta)|\nabla\chi_R|^2
                 +2\chi_R X\cdot\nabla\chi_R\right)|\Psi|^2\\
 &\leq\left(2C_\chi^2+2C_\chi+\frac{\beta C_\chi^2}{2R^2}\right)
          \int_{A_R}|\Psi|^2\longrightarrow0.
\end{split}
\end{equation}
The argument never takes the limit of a quadratic-weight energy without
the cutoff.

By \eqref{eq:removal-moments}, dominated convergence gives
$\int B_j(q_j)|u_R|^2\to\int B_j(q_j)|\Psi|^2\leq0$ for each $j\in J$.
Thus \eqref{eq:removal-quadratic-boundary} controls the term $M_{\beta,U}$,
and the moment bounds control every correction with $\lambda_j\geq0$:
\[
 \limsup_{R\to\infty}
 \left(M_{\beta,U}[u_R]+\sum_{j\in J}\lambda_j\int B_j(q_j)|u_R|^2\right)
 \leq0.
\]
By the assumption of the proposition, the same expression is at least
$\varepsilon\|u_R\|^2\to\varepsilon>0$, a contradiction. Hence no eigenvalue
at or below $T$ exists. The HVZ theorem (Theorem~\ref{thm:hvz}) gives
$\AtomE{N}{Z}\leq T$ and makes any spectral infimum below $T$ an
eigenvalue. It follows that $\AtomE{N}{Z}=T$.
\end{proof}