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\centerline{\bf Failure of Saturated Ferromagnetism}
\centerline{\bf for the Hubbard Model with Two Holes}
\vskip1cm
\centerline{B\'alint T\'oth$^{\star}$}
\hbox{}
\centerline{Department of Mathematics}
\ce
\centerline{Riccarton, EH14 4AS, Scotland}
\vskip1.2cm
\hbox{\centerline{\vbox{\hsize8cm
\noindent
{\bf Abstract.} We consider the Hubbard model on a finite set of sites with
non-positive hopping matrix elements and infinitely strong on-site repulsion.
Nagaoka's theorem states that in this model the relative ground state in the
sector with one unoccupied site is maximally ferromagnetic. We show that this
phenomenon is a consequence of a combinatorial coincidence valid in the one-hole
regime only. In the case of more than one hole there is no reason to ex\-pect
maximally ferromagnetic ground states. We prove this claim for the case of two holes
for models
defined on a class of graphs which contains all tori that are not too small.}}}
\vskip3cm
\noindent
--------------------------------
$$
\align
^{\star}&_{\text{On leave from the Mathematical Institute of the
Hungarian Academy of Sciences,}}\\
&_{\text{Re\'altanoda u. 13-15., Budapest, H--1053}}
\endalign
$$
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\bigskip
\noindent
{\bf 1. Introduction}
\smallskip
\noindent
Let \ $\Cal V=\{1,2,\dots,V\}$ be a finite set and \ ${\left(t_{a,b}\right)}_{(a,b)\in\Cal V\times\Cal V}$
a matrix satisfying
$$
\matrix\format\l\qquad&\l\\
t_{a,b}=t_{b,a}\ge0&(\forall a,b\in\Cal V)\\
t_{a,a}=0&(\forall a\in\Cal V)
\endmatrix
\tag 1.1
$$
We consider the Hubbard model on the set of sites \ $\Cal V$ \ with the (formal) Hamiltonian
$$
H=-\frac12\sum_{x,y\in\Cal V}\sum_{\sigma=\uparrow\downarrow}t_{x,y}\left(c^{+}_{x\sigma}-c^{+}_{y\sigma}\right)
\left(c_{x\sigma}-c_{y\sigma}\right)+\frac U2\sum_{x\in\Cal V}n_x\left(1-n_x\right)
\tag 1.2
$$
with infinitely strong on-site repulsion between the particles: $U=+\infty$. Notice the minus sign
in front of the kinetic energy term, it is of crucial importance in the present setup. Its physical
meaning is that the exchange integrals between the different sites \ $x,y\in\Cal V$ are {\it negative.}
As the pair interaction is spin independent, the Hamiltonian conserves the number of spin-up
and spin-down particles separately. Throughout this paper \ $h,m,n$ \ will denote a triplet of non-negative
integers with the sum \ $h+m+n=V$: \ $m$ \ and \ $n$ \ is the number of spin-up and spin-down particles,
\ $h$ \ is the number of holes (=unoccupied sites). (Due to the hard core repulsion
there are no doubly occupied sites.) We denote by \ $\Cal H_{m,n}$ \ the sector (subspace) with
$m $ spin-up and $n$ spin-down particles and by \ $H_{m,n}$ \ the restriction of \ $H$ to \
$\Cal H_{m,n}$. It is easy to see, that
$$
\Big[[m+n=m'+n']\,\land\,[\max\{m,n\}\ge\max\{m',n'\}]\Big] \Rightarrow\Big[\text{spec} \left(H_{m,n}\right)\subset
\text{spec} \left(H_{m',n'}\right)\Big]
$$
since by a rotation in spin space one can lift isometrically \ $L:\Cal H_{m,n}\to\Cal H_{m',n'}$, changing the
Hamiltonian canonically: $LH_{m,n}=H_{m',n'}L$. In particular we always have
$$
\epsilon_{m+n,0}\ge\epsilon_{m,n}
\tag 1.3
$$
where \ $\epsilon_{m,n}$ \ stands for the ground state energy of \ $H_{m,n}$.
Nagaoka's theorem (see [2], [4], [6]), stated in [5] in its most general form,
says that under the conditions stated above
$$
\big[m+n=V-1\big]\Rightarrow\big[\epsilon_{m+n,0}=\epsilon_{m,n}\big].
\tag 1.4
$$
and if some additional connectivity condition is satisfied (see e.g. [5]), then the ground states
\ $\epsilon_{m,n},\quad m+n=V-1$ \ are nondegenerate. This means that in the \ $V-1$ particle (one hole)
sector\ $\bigoplus_{m+n=V-1}\Cal H_{m,n}$ \ the global ground state is {\it maximally ferromagnetic\,} and this
ground state is unique up to the \ ($V-1$)-fold degeneracy due to rotation in spin space.
The phenomenon was considered interesting, since it might have provided some insight into
the mechanism of itinerant ferromagnetism.
In the present paper we consider the same problem in the case of more than one hole. In the second section
we give a non-standard, alternative representation of the Hamiltonians \ $H_{m,n}$ \ from which clearly
emerges the {\it combinatorial reason\,} why Nagaoka's theorem holds for the case of one hole
and why there is absolutely no reason whatsoever to expect a similar theorem to hold in the case of more
than one hole. The last two sections are devoted to the case $h=2, \ m=1, \ n=V-3$. In section three we
give an upper bound on \ $\epsilon_{1,V-3}$ \ in terms of some more easily handable operator. Using this bound,
in section four we show that \ $\epsilon_{1,V-3}<\epsilon_{0,V-2}$ \
for the Hubbard Hamiltonian (1.2), defined on graphs of the form \ $\Cal T_l\times\Cal R$,
where \ $\Cal T_l$ \ is the one-dimensional torus of length \ $l\ge l_0$ \ and \ $\Cal R$ \ is an {\it arbitrary\,}
graph with first nonzero eigenvalue not less than the first nonzero eigenvalue of \ $\Cal T_l$ .
Consequently in these models the global ground state in the two-hole sector is {\it not\,} maximally ferromagnetic.
Thus we may conclude, that Nagaoka's theorem is the consequence of a combinatorial coincidence valid
in the one-hole regime only and it can't provide a mechanism of itinerant
ferromagnetism. However, let us mention here that an alternative mechanism of
itinerant (anti-) ferromagnetism has been considered recently by Lieb in ref.
[3], where the results and ideas of ref. [1] are further developed.
\vskip2mm
\centerline{* \ * \ *}
\vskip2mm
Some terminology: Given a finite set \ $\tilde{\Cal V}$ \ and a matrix
\ ${\left(t_{a,b}\right)}_{(a,b)\in\tilde{\Cal V}\times\tilde{\Cal V}}$ \ satisfying (1.1) we call the
linear operator
$$
\Delta:\ell_2(\tilde{\Cal V},\bold C)\to\ell_2(\tilde{\Cal V},\bold C),\qquad
-\big[\Delta f\big](x)=\sum_{y\in\tilde{\Cal V}}t_{x,y}\big(f(x)-f(y)\big)
\tag 1.5
$$
a {\it discrete Laplacian}. The reason for calling it so is that \ $\Delta$ \
is the generator of a continuous time random walk on \ $\tilde{\Cal V}$, as the
Laplacian is the generator of Brownian motion in continuous space and the operator
\ $\Delta$ \ has very similar properties to the Laplacian. Discrete Laplacians defined
on various sets \ $\tilde{\Cal V}$ \ will emerge at a later stage. $-\Delta$ \ is always
a positive operator. If unrestricted, its lowest eigenvalue is always $0$, the corresponding eigenspace
is \ $f(x)=\text{const.}$. If the matrix \ ${\left(t_{a,b}\right)}_{(a,b)\in\tilde{\Cal V}\times\tilde{\Cal V}}$ \
is the incidence matrix of a nonoriented graph \ $\tilde{\Cal G}=(\tilde{\Cal V},\tilde{\Cal E})$ \ with vertex set
\ $\tilde{\Cal V}$ \ and edge set \ $\tilde{\Cal E}$, i.e.
$$
t_{a,b}=\left\{\matrix\format\l\qquad&\l\\
1&\text{ if }(a,b)\in\tilde{\Cal E}\\
0&\text{ otherwise}
\endmatrix
\right.
\tag 1.6
$$
then we say that \ $\Delta$ \ is the discrete Laplacian of the graph \ $\tilde{\Cal G}$. In the
concrete examples of the last section discrete Laplacians associated with different graphs will arise.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf 2. An Alternative Representation of the Hamiltonian}
\smallskip
\noindent
The natural representation of the sector with \ $m$ \ spin-up and \
$n$ \ spin-down fermions is
$$
\align
\Cal H_{m,n}=\Big\{f\in\ell_2(\Cal V^m\times\Cal V^n&,\bold C)\,\big|\,\\
\big[\forall \rho\in\Cal P_m, \forall\sigma\in\Cal P_n: \
f(\overline x_{\rho(i)};\overline y_{\sigma(j)})&=
\text{sign}(\rho)\text{sign}(\sigma)f(\overline x_i;\overline y_j)\big] \\
\,\land\,
\big[\big(\exists (i,j)\in\{1,\dots,m\}\times\{1,\dots,n&\}: \ x_i=y_j\big)\Rightarrow
f(\overline x_i;\overline y_j)=0\big]\Big\}
\tag 2.1
\endalign
$$
where \ $\Cal P_r$ \ denotes the group of permutations of \ $r$ \ indices
\ $\{1,2,\dots,r\}$ \ and the shorthand notation \ $\overline x_i=(x_1,\dots,x_m),\
\overline y_j=(y_1,\dots,y_n)$ \ etc. has been used. The antisymmetry condition
is due to the Fermi-Dirac statistics and the second, Dirichlet-type
condition is the precise formulation of the hard core repulsion between the
particles. The Hamiltonian acting on this sector is
$$
\align
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\big[H_{m,n}f\big](\overline x_i;\overline y_j)=-\prod_{i=1}^m\prod_{j=1}^n\Big(1-\delta_{x_i,y_j}\Big)\\
&\left(\sum_{i=1}^m\sum_{\xi\in\Cal V}t_{x_i,\xi}\left(f(x_1,\dots,x_i,\dots,x_m;\overline y_j)-
f(x_1,\dots,\xi,\dots,x_m;\overline y_j)\right)\right.\\
&+\left. \sum_{j=1}^n\sum_{\eta\in\Cal V}t_{y_j,\eta}\left(f(\overline x_i;y_1,\dots,y_j,\dots,y_n)-
f(\overline x_i;y_1,\dots,\eta,\dots,y_n)\right)\right).
\tag 2.2
\endalign
$$
We are going to define now another representation of the same sector and the
Hamiltonian. This new representation might look less natural at the first
glance, but hopefully, at the end of this section the reader will be convinced
that the combinatorial aspects become more transparent.
Denote
$$\align
\Cal V_{h,m,n}=\Big\{&\left(\overline z_k,X,Y\right)\in\Cal V^h\times P(\Cal V,m)\times
P(\Cal V,n)\,\Big|\\
&\left[X\cap Y=\emptyset\right]\,\land\,\left[\forall k\in\{1,\dots,h\}:
z_k\notin X\cup Y\right]\Big\}
\tag 2.3
\endalign
$$
where \ $P(\Cal V,r)$ \ is the set of subsets of $\Cal V$ with cardinality $r$. Let \
$\Cal K_{h,m,n}$ \ be the following Hilbert space:
$$
\Cal K_{h,m,n}=\Big\{\phi\in\ell_2(\Cal V_{h,m,n},\bold C)\,\Big|\,
\left(\forall \pi\in\Cal P_h\right): \phi\left(\overline z_{\pi(k)},X,Y\right)=
\text{sign}(\pi)\phi\left(\overline z_k,X,Y\right)\Big\}.
\tag 2.4
$$
{\sl Remark:\,} Clearly, the notation is redundant a bit: it would be completely
enough to keep only the \ $\overline z_k$ \ and \ $X$ (or, alternatively, the \
$\overline z_k$ \ and \ $Y$) variables. However, for aesthetical reasons we prefer
to keep this notation for the moment. In the next section we shall drop the notation
of the third, redundant variable (e.g. $\Cal V_{h,m,n}$ \ and \ $\Cal K_{h,m,n}$ \ will
be denoted by \ $\Cal V_{h,m}$ \ and \ $\Cal K_{h,m}$ etc.)
The dimensions of the spaces \ $\Cal H_{m,n}$ \ and \ $\Cal K_{h,m,n}$
\ are the same:
$$
\text{dim}\left(\Cal H_{m,n}\right)=\text{dim}\left(\Cal K_{h,m,n}\right)=\frac{V!}{h!m!n!}\,.
$$
Next we define a unitary mapping of \ $\Cal H_{m,n}$ \ onto \ $\Cal K_{h,m,n}$.
Fix once for ever an ordering of the sites in \ $\Cal V$, say \ $1,2,\dots,V$,
and define the function
$$
s:\Cal V^{V}\to\{-1,0,1\},\quad s\left(\xi_1,\dots,\xi_V\right)=\sum_{\pi\in\Cal P_{V}}
\text{sign}(\pi)\prod_{\l=1}^{V}\delta_{\xi_l,\pi(l)}.
\tag 2.5
$$
In plain words: $s\left(\xi_1,\dots,\xi_V\right)$ \ is zero if two different variables
coincide, otherwise it is $+1$ or $-1$ according to the sign of the permutation of the
elements of $\Cal V$. Let the linear operator \ $U:\Cal H_{m,n}\to\Cal K_{h,m,n}$ be defined
by
$$
[Uf](\overline z_k,X,Y)={(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)f(\overline x_i;\overline y_j),
\tag 2.6
$$
where \ $\overline x_i=(x_1,\dots,x_m)$ \ and $\overline y_j=(y_1,\dots,y_n)$ \ are
{\it arbitrary\,} orderings of the elements of the sets $X$, respectively $Y$. It is easy to
check that $U$ is an isometry and its adjoint is \ $U^{*}:\Cal K_{h,m,n}\to\Cal H_{m,n}$, defined
by
$$
\left[U^{*}g\right](\overline x_i;\overline y_j)=\left\{
\matrix\format\l&\l\\
{(h!)}^{1/2}s(\overline z_k,\overline x_i,\overline y_j)g(\overline z_k,X,Y)\quad&\text{if }X\cap Y=\emptyset,\\
0 &\text{if }X\cap Y\not=\emptyset,
\endmatrix\right.
\tag 2.7
$$
where \ $X=\{x_1,\dots,x_m\},\,\,\,Y=\{y_1,\dots,y_n\}$ \ and \ $\overline z_k=(z_1,\dots,z_h)$ \
is an {\it arbitrary\,} ordering of the elements of the set $Z=\Cal V\setminus(X\cup Y)$.
We want to find the new representation
$$
\hat H_{h,m,n}=UH_{m,n}U^{*}
\tag 2.8
$$
of the Hamiltonian.
Define the matrix \ $T:\Cal V_{h,m,n}\times\Cal V_{h,m,n}\to \bold R$ \ as follows:
$$
T_{(\overline z_k,X,Y),(\overline z_k^{\prime},X^{\prime},Y^{\prime})}=\left\{
\matrix\format\l&\l\\
t_{a,b} \qquad &\text{ \ if \ } \bold A \,\land\,(\bold B\,\lor\,\bold C\,\lor\,\bold D) \\
0 &\text{ otherwise}
\endmatrix\right.
\tag 2.9
$$
where the conditions \ $\bold A,\,\,\bold B,\,\,\bold C\text{ and }\bold D$ are the following:
$$
\align
&\bold A:= (\exists k\in\{1,\dots,h\}): \left[z_k=a,z_k^{\prime}=b\right]\,\land\,
\left[(l\not=k)\Rightarrow \left(z_l=z_l^{\prime}\right)\right],\\
&\bold B:=\left[X=X^{\prime}\right]\,\land\,\left[Y\circ Y^{\prime}=\{a,b\}\right],\\
&\bold C:=\left[Y=Y^{\prime}\right]\,\land\,\left[X\circ X^{\prime}=\{a,b\}\right],\\
&\bold D:=\left[X=X^{\prime}\right]\,\land\,\left[Y=Y^{\prime}\right]
\endalign
$$
and the linear operator \ $\Delta_{h,m,n}$ \ on \ $\Cal K_{h,m,n}$:
$$
\align
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\left[\Delta_{h,m,n}\phi\right](\overline z_k,X,Y)=\\
&\sum_{(\overline z_k^{\prime},X^{\prime},Y^{\prime})
\in\Cal V_{h,m,n}}T_{(\overline z_k,X,Y),(\overline z_k^{\prime},X^{\prime},Y^{\prime})}
\Big(\phi(\overline z_k,X,Y)-\phi(\overline z_k^{\prime},X^{\prime},Y^{\prime})\Big).
\tag 2.10
\endalign
$$
The operator \ $\Delta_{h,m,n}$ \ is actually defined on the whole \ $\ell_2(\Cal V_{h,m,n},\bold C)$,
and \ $\Cal K_{h,m,n}\subset\ell_2(\Cal V_{h,m,n},\bold C)$ \ is an invariant subspace of it.
Further, let us denote by \ $I_{h,m,n}$ \ the identity in \ $\Cal K_{h,m,n}$.
\vskip3mm
\noindent
{\bf Proposition 1.}
{\sl
$$
\hat H_{h,m,n}=-\left(\sum_{a,b\in\Cal V}t_{a,b}\right)I_{h,m,n}-\Delta_{h,m,n}
\tag 2.11
$$
}
\vskip3mm
\noindent
{\sl Remarks:\,} Before proving the Proposition let us make some comments on this re\-pre\-sen\-ta\-tion.\newline
(1) The first advantage of the representation (2.11) is principial: we clearly see from it why
Nagaoka's theorem holds in the case of one hole and why there is no reason to expect a
generelization of it for the case of more then one hole. Namely: in the case of one hole the
antisymmetry restriction on the \ $\overline z_k$ \ variables is void, i.e. \ $\Cal K_{1,m,n}=
\ell_2(\Cal V_{1,m,n},\bold C)$, for all \ $m+n=V-1$, and we know that in these cases the bottom of the
spectrum for anyone of the \ $-\Delta_{1,m,n}$ \ is $0$ (this is true for any unrestricted discrete Laplacian).
Thus the ground state of the Hamiltonian in anyone of the sectors \ $\Cal H_{m,n},\ \ m+n=V-1$
has the same energy \ $-\sum_{a,b\in\Cal V}t_{a,b}$. The nondegeneracy (up to rotation in spin space)
of this ground state is a simple consequence of the connectivity condition formulated e.g. in [5].
On the other hand in the case of more then one hole we have to compare the bottom of the
spectra of the operators \ $-\Delta_{h,m,n}$ \ {\it restricted\,} to the proper subspaces
\ $\Cal K_{h,m,n}\subset\ell_2(\Cal V_{h,m,n},\bold C)$. It is easy to see that \
$$
\text{spec}\left({\left.\Delta_{h,0,V-h}\right|}_{\Cal K_{h,0,V-h}}\right)\subset
\text{spec}\left({\left.\Delta_{h,m,n}\right|}_{\Cal K_{h,m,n}}\right),\qquad(\forall m,n): m+n=V-h
$$
(see the lifting operator \ $L$ \ defined in (3.15)), but we have no more reason for the coincidence of the
bottom of the spectra. We also understand now why is the sign of the hopping matrix elements so crucial
in Nagaoka's theorem: the bottom of the spectrum of \ $-\Delta$ \ is zero for any unrestricted discrete Laplacian,
but there is no reason for coincidence of the top of the spectra.\newline
(2) The technical advantage will become clear in the next section: in the case of few holes the
combinatorial aspects are more transparent in this representation.
\vskip2mm
\noindent
{\bf Proof:} If \ $z_k=z_l$ \ for some \ $k\not=l$, then clearly \ $\left[\hat H_{h,m,n}g\right]
(\overline z_k,X,Y)=0$ \ on both sides of (2.11). So let's assume \ $z_k\not=z_l$ \ for \ $k\not=l$.
$$
\align
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left[\hat H_{h,m,n}g\right](\overline z_k,X,Y)=-{(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)\\
&\left(\sum_{i=1}^m\sum_{\xi\in\Cal V}t_{x_i,\xi}\left(\left[U^{*}g\right]
(x_1,\dots,x_i,\dots,x_m;\overline y_j)-
\left[U^{*}g\right](x_1,\dots,\xi,\dots,x_m;\overline y_j)\right)\right.\\
&+\left. \sum_{j=1}^n\sum_{\eta\in\Cal V}t_{y_j,\eta}\left(\left[U^{*}g\right]
(\overline x_i;y_1,\dots,y_j,\dots,y_n)-
\left[U^{*}g\right](\overline x_i;y_1,\dots,\eta,\dots,y_n)\right)\right)
\tag 2.12
\endalign
$$
by definition. Note, that since \ $X\cap Y=\emptyset$, \
$\prod_{i=1}^m\prod_{j=1}^n\left(1-\delta_{x_i,y_j}\right)=1$ \ automatically.
As the $z$-variables are assumed to be all different, if we denote $\{z_1,\dots,z_h\}=Z$,
then $X,Y,Z$ is a partition of $\Cal V$ and we can write
$$
\align
&\sum_{i=1}^m\sum_{\xi\in\Cal V}t_{x_i,\xi}\Big(\left[U^{*}g\right](x_1,\dots,x_i,\dots,x_m;\overline y_j)
-\left[U^{*}g\right](x_1,\dots,\xi,\dots,x_m;\overline y_j)\Big)=\\
&\sum_{i=1}^m\left(\!\sum_{\xi\in X}\!+\!\sum_{\xi\in Y}\!+\!\sum_{\xi\in Z}\!\right)t_{x_i,\xi}
\Big(\!\left[U^{*}g\right](x_1,\dots,x_i,\dots,x_m;\overline y_j)-\left[U^{*}g\right](x_1,\dots,
\xi,\dots,x_m;\overline y_j)\!\Big)\!=\\
&\sum_{i=1}^m\left(\sum_{i'=1}^m t_{x_i,x_{i'}}+\sum_{j=1}^n t_{x_i,y_j}\right)\left[U^{*}g\right]
(\overline x_i;\overline y_j)+\\
&\sum_{i=1}^m\sum_{k=1}^h t_{x_i,z_k}\Big(\left[U^{*}g\right](x_1,\dots,x_i,\dots,x_m;\overline y_j)-
\left[U^{*}g\right](x_1,\dots,z_k,\dots,x_m;\overline y_j)\Big)=\\
&\sum_{i=1}^m\left(\sum_{i'=1}^m t_{x_i,x_{i'}}+\sum_{j=1}^n t_{x_i,y_j}+2\sum_{k=1}^ht_{x_i,z_k}\right)
\left[U^{*}g\right](\overline x_i;\overline y_j)-\\
&\sum_{i=1}^m\sum_{k=1}^h t_{x_i,z_k}\Big(\left[U^{*}g\right](x_1,\dots,x_i,\dots,x_m;\overline y_j)
+\left[U^{*}g\right](x_1,\dots,z_k,\dots,x_m;\overline y_j)\Big)
\tag 2.13
\endalign
$$
In the second step we used the fact that for \ $\xi\in(X\setminus\{x_i\})\cup Y$ \
$\left[U^{*}g\right](x_1,\dots,\xi,\dots,x_m;\overline y_j)$ $=0$ \ if $\xi$ is in the position
of $x_i$. In similar way we get
\nopagebreak
$$
\align
&\sum_{j=1}^n\sum_{\eta\in\Cal V}t_{y_j,\eta}\Big(\left[U^{*}g\right](\overline x_i;y_1,\dots,y_j,\dots,y_n)
-\left[U^{*}g\right](\overline x_i;y_1,\dots,\eta,\dots,y_n)\Big)=\\
&\sum_{j=1}^n\left(\sum_{j'=1}^n t_{y_j,y_{j'}}+\sum_{i=1}^m t_{y_j,x_i}+2\sum_{k=1}^ht_{y_j,z_k}\right)
\left[U^{*}g\right](\overline x_i;\overline y_j)-\\
&\sum_{j=1}^n\sum_{k=1}^h t_{y_j,z_k}\Big(\left[U^{*}g\right](\overline x_i;y_1,\dots,y_j,\dots,y_n)
+\left[U^{*}g\right](\overline x_i;y_1,\dots,z_k,\dots,y_n)\Big)
\tag 2.14
\endalign
$$
Pluging~(2.13),~(2.14)~in~(2.12),~further,~adding~and~subtracting
\nopagebreak
$$
\sum_{k=1}^h\sum_{k'=1}^ht_{z_k,z_{k'}}\left[U^{*}g\right](\overline x_i;\overline y_j)
$$
we get
$$
\align
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left[\hat H_{h,m,n}g\right](\overline z_k,X,Y)=\\
&\ \ {(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)
\left(-\sum_{a,b\in\Cal V}t_{a,b}\left[U^{*}g\right](\overline x_i;\overline y_j)
+\sum_{k=1}^h\sum_{k'=1}^h t_{z_k,z_{k'}}\left[U^{*}g\right](\overline x_i;\overline y_j)
\right.\\
&+\sum_{i=1}^m\sum_{k=1}^h t_{x_i,z_k}\Big(\left[U^{*}g\right](x_1,\dots,x_i,\dots,x_m;\overline y_j)
+\left[U^{*}g\right](x_1,\dots,z_k,\dots,x_m;\overline y_j)\Big)\\
&+\left.\sum_{j=1}^n\sum_{k=1}^h t_{y_j,z_k}\Big(\left[U^{*}g\right](\overline x_i;y_1,\dots,y_j,\dots,y_n)
+\left[U^{*}g\right](\overline x_i;y_1,\dots,z_k,\dots,y_n)\Big)\right)
\tag 2.15
\endalign
$$
Note that
$$
\align
{(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)\left[U^{*}g\right](\overline x_i;\overline y_j)&=
g(\overline z_k,X,Y)\\
{(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)\left[U^{*}g\right](x_1,\dots,z_k,\dots,x_m;\overline y_j)&=
-g(z_1,\dots,x_i,\dots,z_h,X\circ\{x_i,z_k\},Y)\\
{(h!)}^{-1/2}s(\overline z_k,\overline x_i,\overline y_j)\left[U^{*}g\right](\overline x_i;y_1,\dots,z_k,\dots,y_n)&=
-g(z_1,\dots,y_j,\dots,z_h,X,Y\circ\{y_j,z_k\})
\endalign
$$
where \ $X=\{x_1,\dots,x_m\},\ \ Y=\{y_1,\dots,y_n\}$, in the second (third) equality
on the left hand side $z_k$ replaces $x_i$ ($y_j$) and on the right hand side
$x_i$ ($y_j$) replaces $z_k$. Using these equalities finally we get
$$
\align
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left[\hat H_{h,m,n}g\right](\overline z_k,X,Y)=-\left(\sum_{a,b\in\Cal V}t_{a,b}\right)g(\overline z_k,X,Y)+\\
&\left[\sum_{k=1}^h\sum_{x\in X}t_{z_k,x}\Big(g(z_1,\dots,z_k,\dots,z_h,X,Y)-g(z_1,\dots,x,\dots,z_h,X\circ
\{x,z_k\},Y)\Big)+\right.\\
&\ \ \sum_{k=1}^h\sum_{y\in Y}t_{z_k,y}\Big(g(z_1,\dots,z_k,\dots,z_h,X,Y)-g(z_1,\dots,y,\dots,z_h,X,Y\circ
\{y,z_k\})\Big)+\\
&\left.\ \ \sum_{k=1}^h\sum_{k'=1}^h\!t_{z_k,z_{k'}}\!\Big(g(z_1,...,z_k,...,z_{k'},...,z_h,X,Y)\!-\!
g(z_1,...,z_{k'},...,z_{k'},...,z_h,X,Y)\Big)\!\!\right]
\tag 2.16
\endalign
$$
(2.16) is exactly the expanded form of (2.11), thus Proposition 1. is proved.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf 3. h=2, m=1, n=V--3: Preparation}
\smallskip
\noindent
The rest of this paper is devoted to the case \ $h=2,\ \ m=1,\ \ n=V-3$,
i.e. two holes and total spin in the z-direction one less than the possible maximum.
More exactly we want to find conditions under which the lowest eigenvalue of \
$-\Delta_{2,1,V-3}$ \ restricted to \ $\Cal K_{2,1,V-3}$ \ is strictly less than the lowest
eigenvalue of \ $-\Delta_{2,0,V-2}$ \ restricted to \ $\Cal K_{2,0,V-2}$. At this stage
introduction of a less general notation is convenient. Namaly, as pointed out in the
remark after (2.4), we are allowed to drop the notation of the third, redundant variable ($Y$).
As we are going to use a couple of different vector spaces and discrete Laplacians, let
us define them from start. The discrete sets on which the functions of ``different
level'' are defined, are the following:
$$
\align
\Cal V&=\{1,2,\dots,V\}
\tag 3.1\\
\Cal V_{2,0}&=\left\{(z_1,z_2)\,\right|\left.\,z_1,z_2\in\Cal V\right\}
\tag 3.2\\
\Cal V_{2,1}&=\left\{(z_1,z_2|x)\,\right|\left.\,z_1,z_2,x\in\Cal V,\ \ z_1\not=x\not=z_2\right\}
\tag 3.3\\
\Cal V_{2\not=}&=\left\{(z,x)\,\right|\left.\,z,x\in\Cal V,\ \ z\not=x\right\}
\tag 3.4
\endalign
$$
The corresponding Hilbert spaces are
$$
\align
\Cal K&=\ell_2(\Cal V,\bold C)
\tag 3.5\\
\Cal K_{2,0}&=\left\{\phi\in\ell_2(\Cal V_{2,0},\bold C)\,\right|\left.\,\left(\forall (z_1,z_2)\in\Cal V_{2,0}\right):
\phi(z_1,z_2)=-\phi(z_2,z_1)\right\}
\tag 3.6\\
\Cal K_{2,1}&=\left\{\phi\in\ell_2(\Cal V_{2,1},\bold C)\,\right|\left.\,\left(\forall (z_1,z_2|x)\in\Cal V_{2,1}\right):
\phi(z_1,z_2|x)=-\phi(z_2,z_1|x)\right\}
\tag 3.7\\
\Cal K_{2\not=}&=\big\{\varphi\in\ell_2(\Cal V_{2\not=}\,\bold C)\,\big|\,
[\left(\forall (z,x)\!\in\!\Cal V_{2\not=}\right)\!:
\varphi(z,x)\!=\!\varphi(x,z)]\,\land \,[\left(\forall z\!\in\!\Cal V\right)\!:\!\!\!\!\!\sum_{x\in\Cal V\setminus
\{z\}}\!\!\!\!\!\varphi(z,x)\!=\!0]\big\}
\tag 3.8
\endalign
$$
The different discrete Laplacians
$$
\Delta_{\sharp}:\Cal K_{\sharp}\to\Cal K_{\sharp},
$$
where \ $\sharp$ \ stands for any one of the subscripts, are defined as follows:
$$
\align
\left[-\Delta f\right](z)&=\sum_{\zeta\in\Cal V}t_{z,\zeta}\big(f(z)-f(\zeta)\big)
\tag 3.9\\
\left[-\Delta_{2,0} \phi\right](z_1,z_2)&=\sum_{\zeta\in\Cal V}t_{z_1,\zeta}\big(\phi(z_1,z_2)\!-\!\phi(\zeta,z_2)\big)
\!+\!\sum_{\zeta\in\Cal V}t_{z_2,\zeta}\big(\phi(z_1,z_2)\!-\!\phi(z_1,\zeta)\big)
\tag 3.10\\
\left[-\Delta_{2,1} \phi\right](z_1,z_2|x)&=\!\!\!\!\sum_{\zeta\in\Cal V\setminus\{x\}}\!\!\!\!t_{z_1,\zeta}
\big(\phi(z_1,z_2|x)\!-\!\phi(\zeta,z_2|x)\big)
\!+\!\!\!\!\!\!\!\sum_{\zeta\in\Cal V\setminus\{x\}}\!\!\!\!\!\!t_{z_2,\zeta}\big(\phi(z_1,z_2|x)\!-\!\phi(z_1,\zeta|x)\big)\!\\
+t_{z_1,x}\big(&\phi(z_1,z_2|x)-\phi(x,z_2|z_1)\big)+t_{z_2,x}\big(\phi(z_1,z_2|x)-\phi(z_1,x|z_2)\big)
\tag 3.11\\
\left[-\Delta_{2\not=} \varphi\right](z,x)&=\sum_{\zeta\in\Cal V\setminus\{x\}}t_{z,\zeta}
\big(\varphi(z,x)-\varphi(\zeta,x)\big)+\sum_{\xi\in\Cal V\setminus\{z\}}t_{x,\xi}
\big(\varphi(z,x)-\varphi(z,\xi)\big)
\tag 3.12
\endalign
$$
{\sl Remark:\,} The operators \ $\Delta_{\sharp}$ \ are actually defined on the whole
\ $\ell_2(\Cal V_{\sharp},\bold C)$ \ and \ $\Cal K_{\sharp}\subset\ell_2(\Cal V_{\sharp},\bold C)$ \ are invariant
subspaces of \ $\Delta_{\sharp}$. It is clear that \ $\Cal K_{2,0},\ \Cal K_{2,1},\ \Delta_{2,0}\text{ and }\Delta_{2,1}$ \
are the same as \ $\Cal K_{2,0,V-2},\ \Cal K_{2,1,V-3},\ \Delta_{2,0,V-2}\text{ and }\Delta_{2,1,V-3}$ \ of the
previous section.
The bottom of the spectrum of \ $-\Delta_{\sharp}, \ \ \sharp=(2,0),\ (2,1),\ (2\!\!\not=)$ \ is given
by the variational formula:
$$
\lambda^{*}_{\sharp}=\inf_{\phi\in\Cal K_{\sharp}}\frac{{\big\langle\phi,-\Delta_{\sharp}\phi\big\rangle}_{\sharp}}
{{\big\langle\phi,\phi\big\rangle}_{\sharp}}
\tag 3.13
$$
and we trivially have
$$
\lambda^{*}_{2,0}=\lambda^{*}=\inf\left\{\frac{\big\langle f,-\Delta f\big\rangle}{\big\langle f,f\big\rangle}\,\Big|\,
f\in\ell_2(\Cal V,\bold C),\ \ \ \sum_{z\in\Cal V}f(z)=0\right\}.
\tag 3.14
$$
There is a natural lifting of \ $\Cal K_{2,0}$ \ to \ $\Cal K_{2,1}$:
$$
L:\Cal K_{2,0}\to\Cal K_{2,1},\qquad \big[L\phi\big](z_1,z_2|x)=\phi(z_1,z_2)
\tag 3.15
$$
such that
$$
L\Delta_{2,0}=\Delta_{2,1}L.
\tag 3.16
$$
>From this equality follows that the spectrum of \ $\Delta_{2,0}$ \ is contained in the spectrum of
\ $\Delta_{2,1}$ \ and if we want to find an eigenvalue of \ $\Delta_{2,1}$ \ not belonging
to the spectrum of \ $\Delta_{2,0}$, then we have to concentrate on the subspace
$$
\text{Ran}(L)^{\perp}=\Big\{\phi\in\Cal K_{2,1}\,\Big|\,\left(\forall z_1,z_2\in\Cal V\right):
\sum_{x\in\Cal V\setminus\{z_1,z_2\}}\phi(z_1,z_2|x)=0\Big\}.
\tag 3.17
$$
On the other hand, knowing that the eigenfunction belonging to the lowest eigenvalue
of \ $-\Delta_{2,0}$ \ is of the form \ $\phi(z_1,z_2)=\varphi(z_1)-\varphi(z_2)$, it is
not completely out of the blue to try to minimize \
${\big\langle\phi,-\Delta_{2,1}\phi\big\rangle}_{2,1}\big/{\big\langle\phi,\phi\big\rangle}_{2,1}$
on the subspace
$$
\Cal L=\left\{\phi\in\Cal K_{2,1}\,\big|\,\phi(z_1,z_2|x)=\varphi(z_1,x)-\varphi(z_2,x)\right\},
\tag 3.18
$$
and after some straightforward calculations we find that the intersection of the
subspaces \ $\text{Ran}(L)^{\perp}$ \ and \ $\Cal L$ \ is
$$
\text{Ran}(L)^{\perp}\cap\Cal L=\text{Ran}(M)
\tag 3.19
$$
where \ $M$ \ is the linear operator
$$
M:\Cal K_{2\not=}\to\Cal K_{2,1},\qquad \big[M\varphi\big](z_1,z_2|x)=\varphi(z_1,x)-\varphi(z_2,x).
\tag 3.20
$$
That is how the space \ $\Cal K_{2\not=}$ \ arises.
Next we minimize \
${\big\langle\phi,-\Delta_{2,1}\phi\big\rangle}_{2,1}\big/{\big\langle\phi,\phi\big\rangle}_{2,1}$
on the subspace \ $\text{Ran}(M)$. Straightforward calculations show that for \ $\varphi\in\Cal K_{2\not=}$
$$
{\big\langle M\varphi,M\varphi\big\rangle}_{2,1}=2\left(V-1\right){\big\langle \varphi,\varphi\big\rangle}_{2\not=}
\tag 3.21
$$
and
$$
{\big\langle M\varphi,-\Delta_{2,1}M\varphi\big\rangle}_{2,1}=
V{\big\langle \varphi,-\Delta_{2\not=}\varphi\big\rangle}_{2\not=}.
\tag 3.22
$$
So we may conclude this section with the following
\vskip3mm
\noindent
{\bf Proposition 2.}
{\sl
$$
\lambda^{*}_{2,1}\le\frac{V}{2\left(V-1\right)}\lambda^{*}_{2\not=}.
\tag 3.23
$$
}
\vskip3mm
\noindent
{\sl Remark:\,} The subspace \ $\text{Ran}(M)$ \ in general is {\it not\,} an invariant subspace of
\ $\Delta_{2,1}$, so the right hand side of (3.23) is not necessarily an eigenvalue of \ $-\Delta_{2,1}$,
just an upper bound on the bottom of its spectrum. Clearly \ $\Delta_{2\not=}$ \ is much easier to
handle than \ $\Delta_{2,1}$.
Exploiting this bound, in the next section we shall prove for a collection of examples
that \ $\lambda^{*}_{2,1}<\lambda^{*}_{2,0}$ by showing that \ $\frac{V}{2\left(V-1\right)}\lambda^{*}_{2\not=}<
\lambda^{*}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip8mm
\noindent
{\bf 4. h=2, m=1, n=V--3: Concrete Examples}
\smallskip
\noindent
In our concrete examples \ ${\left(t_{a,b}\right)}_{(a,b)\in\Cal V\times\Cal V}$ \
will be the incidence matrix of a graph \ $\Cal G=(\Cal V,\Cal E)$ \ and, consequently,
$\Delta$ will be the discrete Laplacian of \ $\Cal G$. The graph \ $\Cal G_{2\not=}$
is obtained from \ $\Cal G\times\Cal G$ \ by removing all its diagonal vertices and all
the edges joining them to the rest of \ $\Cal G\times\Cal G$:
$\Cal G_{2\not=}=(\Cal V_{2\not=},\Cal E_{2\not=})$, where \ $\Cal V_{2\not=}$ \ is
defined in (3.4) and
$$
\Cal E_{2\not=}=\Big\{\big((x,z),(x',z')\big)\in\Cal V_{2\not=}\times\Cal V_{2\not=}\,\Big|
\, \big[x=x'\land (z,z')\in\Cal E\big]\lor\big[z=z'\land (x,x')\in\Cal E\big]\Big\}.
\tag 4.1
$$
$\Delta_{2\not=}$ \ is the discrete Laplacian associated to this graph. $\Cal K_{2\not=}$,
as defined in (3.8), is an invariant subspace of \ $\Delta_{2\not=}$ \ and \ $\lambda^*_{2\not=}$ \
is the smallest eigenvalue of \ $\Delta_{2\not=}\Big|_{\Cal K_{2\not=}}$. We are going to prove
:
$$
\frac{V}{2(V-1)}\lambda^*_{2\not=}<\lambda^*,
\tag 4.2
$$
for a class of graphs, which include as particular case discrete tori in any dimensions.
\vskip1mm
\noindent
Let \ $\Cal T_{{l}}$ \ be the one dimensional discrete torus of length \ ${l}$.
Two vertices $x,y\!\in\!\{\!0\!,\!1\!,\!\dots\!,\!{l\!-\!1}\!\}$ are joined by an edge of \ $\Cal T_{{l}}$ \
if $(|x-y| \ \text{mod} \ {l})=1,l-1$. Our class of examples will consists of graphs of the form
$$
\Cal G=\Cal T_{{l}}\times \Cal R
$$
where \ $\Cal R$ \ is an arbitrary finite graph on \ $r$ \ vertices.
We prove the following
\vskip3mm
\noindent
{\bf Proposition 3.}
{\sl There exists an $l_0<\infty$ such that for any $l>l_0$ and any
finite graph $\Cal R$
$$
\frac{V}{2(V-1)}\lambda^*_{2\not=}<2\left(1-\cos\frac{2\pi}{{l}}\right)=\lambda.
\tag 4.3
$$
}
\vskip3mm
\noindent
{\sl Remarks:\,} In (4.3) \ $V={l}r$, of course and the right hand is \
$\lambda=\lambda^*(\Cal T_{{l}})$. Clearly \
$\lambda^*(\Cal G)=\lambda^*(\Cal T_{{l}})\land\lambda^*(\Cal R)$,
so if \ $\lambda^*(\Cal T_{{l}})\le \lambda^*(\Cal R)$,
then (4.3) is equivalent to (4.2), and our main
claim is proved. In particular, discrete tori in any dimensions with longest side longer than \ $l_0$ \ are included.
(Recall
that \ $\lambda^*(\Cal G)$ \ denotes the smalest positive (non-zero) eigenvalue of \ $-\Delta$ \
associated to the graph \ $\Cal G$.)
\noindent
{\bf Proof:} In the present proof we adopt the following notation: for \ $x,y\in\{0,1,\dots,{l-1}\}$ \
$x\pm y$ \ will be understood as \ $(x\pm y \ \text{mod} \ {l})$. The (real valued) functions \
$f,g,\dots$ \ will be defined on \ $\{1,\dots,{l-1}\}$, their scalar product and norm is defined as
$$
(f,g)=\sum_{z=1}^{{l-1}}f(z)g(z),\qquad ||g||^2=(g,g)
\tag 4.4
$$
Beware of the lower limit of the summation! Vertices of the graph \ $\Cal R$ \ will be denoted by \ $\alpha, \beta,\dots$.
Denote
$$
f:\{1,2,\dots {l-1}\}\to\bold R\qquad f(z)=\cos\frac{2\pi}{{l}}z.
\tag 4.5
$$
We minimize \
$\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{2\not=}/\left\langle\phi,\phi\right\rangle_{2\not=}$ \
among trial functions of the form
$$
\phi(x,\alpha;y,\beta)=\sqrt{\frac{2}{r^2l}}\left\{\matrix\format\l&\l\\
\dsize\cos\left(\frac{2\pi}{{l}}(x+y)\right)g(x-y)\qquad&\text{ if } x\not=y\\
\dsize-\frac{r}{r-1}(f,g) &\text{ if } x=y\text{ and } \alpha\not=\beta
\endmatrix\right.
\tag 4.6
$$
where \ $g$ \ is an even function on \ $\{1,2,\dots,{l-1}\}$:
$$
g(z)=g({l}-z),\qquad z\in\{1,2,\dots {l-1}\}
\tag 4.7
$$
and in the case \ $r=1$ \ (i.e. when \ $\Cal G=\Cal T_{{l}}$) the extra condition
$$
(f,g)=\sum_{z=1}^{{l-1}}\cos\left(\frac{2\pi}{{l}}z\right)g(z)=0
\tag 4.8
$$
is imposed. (Notice, that the \ $z=0$ \ term is missing from the sum.)
One can easily check, that the conditions (3.8) for \ $\phi\in\Cal K_{2\not=}$ \ are satisfied.
After some straightforward calculations we find the $\Cal K_{2\not=}$-norm of \ $\phi$
$$
\align
\text{for }r>1:\qquad&||\phi||^2_{2\not=}=\left[||g||^2+\frac{r}{r-1}(f,g)^2\right]
\tag 4.9\\
\text{for }r=1:\qquad&||\phi||^2_{2\not=}=||g||^2
\tag 4.9$^{\prime}$
\endalign
$$
and the Dirichlet form
$$
\align
\text{for }r>1:\qquad&\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{2\not=}=\\
\!\!\!\!\!2\Bigg[\!\!(g,Sg)\!\!+\!\!
\frac{2}{r(r-1)}\!\!&\left(\!\!r^2(f,g)^2\!\!-\!\!2r(r-1)\cos\left(\frac{2\pi}{{l}}\right)(f,g)g(1)\!\!+\!\!
(r-1)^2g^2\!(1)\right)\!\!\Bigg]
\tag 4.10\\
\text{for }r=1:\qquad&\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{2\not=}=2(g,Sg)
\tag 4.10$^{\prime}$
\endalign
$$
where \ $S$ \ is the linear operator on \ $\ell_2(\{1,2,\dots,{l-1}\})$ \ defined
below
$$
[Sg](z)=\left\{\matrix\format\r&\l\qquad&\l\\
&g(z)-\cos\left(\dsize\frac{2\pi}{{l}}\right)g(z+1) &\text{ if }\quad z=1\\
2&g(z)-\cos\left(\dsize\frac{2\pi}{{l}}\right)\Big(g(z+1)+g(z-1)\Big) &\text{ if }\quad z\not=1,{l-1}\\
&g(z)-\cos\left(\dsize\frac{2\pi}{{l}}\right)g(z-1) &\text{ if }\quad z={l-1}
\endmatrix
\right.
\tag 4.11
$$
Let's consider first the case \ $r=1$ (i.e. \ $\Cal G=\Cal T_{{l}}$).
The smallest eigenvalue of \ $S$ \ on the subspace of even \ $g$-s satisfying (4.7)
is found to be
$$
\overline{\lambda}=2\left(1-\cos\frac{2\pi}{{l}}\cosh\theta\right)
\tag 4.12
$$
where \ $\pm\theta$ \ are the only real solutions of the equation
$$
e^{{l}\theta}=\frac{e^\theta-\cos\frac{2\pi}{{l}}}{\cos\frac{2\pi}{{l}}-e^{-\theta}}.
\tag 4.13
$$
The corresponding eigenfunction of \ $S$ \ is
$$
\overline g(z)=\cosh\left[\theta\left(\frac{{l}}{2}-z\right)\right],\qquad z\in\{1,2,\dots,{l-1}\}
\tag 4.14
$$
and one can check that \ $\overline g$ \ satisfies (4.8).
For large values of \ $l$ \ the asymptotic behaviour of \ $\lambda$ \ is
$$
\lambda=2\left(1-\cos\frac{2\pi}{{l}}\right)=\frac{4\pi^2}{l^2}+o(l^{-3})
\tag 4.15
$$
and solving (4.13) up to the first three leading terms after some tedious, but
straightforward calculations we get
$$
\theta=\frac{2\pi}{l^{3/2}}+\frac{\pi^2(\pi+3)}{3l^{5/2}}+o(l^{-5/2})
\tag 4.16
$$
and
$$
\lambda-\overline\lambda=2(\cosh\theta-1)\cos\frac{2\pi}{{l}}=
\frac{4\pi^2}{l^{3}}+\frac{4\pi^3(\pi+3)}{3l^{4}}+o(l^{-4}).
\tag 4.17
$$
Using these asymptotic expansions and (4.9$^{\prime}$), (4.10$^{\prime}$) we finally get
$$
(V-1)\lambda-\frac{V}{2}\frac
{\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{\!2\not=}}{\left\langle\phi,\phi\right\rangle_{2\not=}}=
{l}\left(\lambda-\overline\lambda\right)-\lambda=\frac{4\pi^3(\pi+3)}{3l^{3}}+o(l^{-3}).
\tag 4.18
$$
This expression is positive for sufficiently large values of \ $l$, which proves our Proposition~3. for
\ $r=1$.
We turn now to the case \ $r>1$. Choose
$$
g=\overline g - c1\!\!1
\tag 4.19
$$
where \ $\overline g$ \ is the eigenfunction of \ $S$ \ defined in (4.14), \ $c$ \ is
a real number and \ $1\!\!1$ \ is the function identically one on \ $\{1,2,\dots,{l-1}\}$.
With this choice formulae (4.9) and (4.10) become
$$
\align
||\phi||^2_{2\not=}&=
\left[||\overline g||^2-2c\left(\overline g,1\!\!1\right)
+c^2\left({l}+\frac{1}{r-1}\right)\right]
\tag 4.9$^{\prime\prime}$\\
\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{2\not=}&=2
\Bigg[\overline\lambda||\overline g||^2-2\overline\lambda c\left(\overline g,1\!\!1\right)
+ c^2\left(1\!\!1,S1\!\!1\right)+
\tag 4.10$^{\prime\prime}$\\
\frac{2}{r(r+1)}&\!\!\left(r^2c^2\!+\!2\cos\left(\frac{2\pi}{{l}}\right)r(r-1)c(\overline g(1)+c)
\!+\!(r-1)^2(\overline g(1)+c)^2\right)\!\!\Bigg]
\endalign
$$
This last formula simplifies considerably with the choice
$$
c=\frac{r-1}{2r-1}\overline g(1).
\tag 4.20
$$
Noting that
$$
\left(1\!\!1,S1\!\!1\right)=({l}-2)\lambda
\tag 2.21
$$
we finally get
$$
\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{2\not=}=2
\left[\overline\lambda||\overline g||^2-2\overline\lambda c\left(\overline g,1\!\!1\right)
+\lambda c^2\left({l}+\frac{1}{r-1}\right)\right].
\tag 4.22
$$
Using the asymptotics (4.15), (4.17) (to first leading terms only) and the following straightforward
expansions:
$$
\align
||\overline g||^2&={l}+o(l)=\left(\overline g,1\!\!1\right)
\tag 4.23\\
c&=\frac{r-1}{2r-1}+o(1)
\tag 4.24
\endalign
$$
we finally get
$$
\align
2(V-1)\lambda\left\langle\phi,\phi\right\rangle_{2\not=}-
V\left\langle\phi,-\Delta_{2\not=}\phi\right\rangle_{\!2\not=}&=\\
\left[||\overline g||^2-2c\left(\overline g,1\!\!1\right)\right]
\left[{l}r\left(\lambda-\overline\lambda\right)-\lambda\right]&-
c^2\left({l}+\frac{1}{r-1}\right)\lambda=\\
\frac{r(r-1)}{(2r-1)^2}\frac{4\pi^2}{l}+o(l^{-1})
\tag 4.25
\endalign
$$
Since the coefficient of the leading term in (4.25) is positive, this expression is eventually positive,
which proves Proposition 3. for the cases \ $r>1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1cm
\noindent
{\bf Acknowledgements.} It is a pleasure to thank Oliver Penrose for many illuminating
discussions on the subject. This work was supported by the British Science and
Engineering Research Council. \newline
During the preparation of this manuscript I have learnt about Refs. [7, 8], where
similar results are obtained by different methods. I wish to thank E.H. Lieb for
drawing my attention to these papers and for other helpful comments.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf References}
\smallskip
\noindent
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Fourteenth Solvay Conference.\,} Wiley Interscience, New York
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\item{ [3] } E.\,H. Lieb: Two theorems on the Hubbard Model. {\sl Phys. Rev.
Letters.\,} {\bf 62}, 1201-1204 (1989)
\item{ [4] } Y. Nagaoka: Ferromagnetism in a Narrow Almost Half Filled
$s$-Band.
{\sl Phys. Rev.\,} {\bf 147}, 392-405 (1966)
\item{ [5] } H. Tasaki: Extension of Nagaoka's Theorem on the Large-$U$ Hubbard Model.
{Phys. Rev. B.\,} {\bf 40}, 9192-9193 (1989)
\item{ [6] } D.\,J. Thouless: Exchange in Solid $^3$He and the Heisenberg
Hamiltonian. {\sl Proc. Phys. Soc. London.\,} {\bf 86}, 893-904
(1965)
\item{ [7] } B. Doucot, X.\,G. Wen: Instability of the Nagaoka State with More Than One Hole.
{\sl Phys. Rev. B\,} {\bf 40}, 2719-2722 (1989)
\item{ [8] } A. S\"ut\H o: Absence of Highest-Spin Ground States in the Hubbard Model. (submitted to
Commun. Math. Phys.) (1991)
\end
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ENDBODY