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\title{ {\it Erratum}\\
Exact Ground State Energy of the Strong-Coupling
Polaron\\
{\small Commun.~Math.~Phys.~{\bf 183}, 511-519, (1997)}}
\author{Elliott~H.~Lieb$^1$ and Lawrence~E.~Thomas$^2$\\
\footnotesize \it $^1$Departments of Physics and Mathematics,
Jadwin Hall, Princeton University,\\
\footnotesize \it P.~O.~Box 708, Princeton, New Jersey 08544\\
\footnotesize \it
$^2$Department of Mathematics, University of Virginia,
Charlottesville, Virginia 22903}
\maketitle
We are grateful to Professor Andrey V. Soldatov of the Moscow Steklov
Mathematical Institute for calling our attention to an error in our
paper. The commutator inequality (8) in our step {\bf I},
namely $|k_j||\langle a_{\bf
k}e^{i{\bf k}\cdot{\bf x}}\rangle|\leq 2\langle
p_j^2\rangle^{1/2}\langle a_{\bf k}^*a_{\bf k}^{\phantom
*}\rangle^{1/2}$, is not correct. Rather, the right side
of this inequality should be $\langle p_j^2\rangle^{1/2}(\langle
a_{\bf k}^*a_{\bf k}^{\phantom *}\rangle^{1/2}+\langle a_{\bf
k}^{\phantom*}a_{\bf k}^{*}\rangle^{1/2})$ or a related expression.
The extra factor
$\langle a_{\bf k}^{\phantom*}a_{\bf k}^{*}\rangle^{1/2}$ with the
$a_{\bf k}^{\phantom*}$ and $a_{\bf k}^{*}$ not in normal order
generates uncontrolled mischief with, for example, the right side of
the ultraviolet bound (10) containing an additional term $\sum_{|{\bf
k}| \geq K}1/2 = \infty$.
The situation is easily remedied with the help of the method introduced
by Lieb and Yamazaki [14] to obtain the previous rigorous
lower bound on the polaron energy. Our main result, (31), is still valid. Indeed, it is improved slightly!
Define the (vector) operator ${\bf Z}=
(Z_1, Z_2, Z_3)$ with components
\begin{equation}
Z_j= (\frac{4\pi\alpha}{V})^{1/2}\sum_{|{\bf k}|\geq
K} k_j \frac{{a}_{\bf k}}{|{\bf k}|^3}e^{i{\bf k}\cdot{\bf
x}},\hspace{.2 in} j= 1,2,3.
\end{equation}
Then the commutator estimate (8) is replaced by
\begin{eqnarray}
-(\frac{4\pi\alpha}{V})^{1/2}\sum_{|{\bf k}|\geq
K}\left[\langle\frac{{a}_{\bf k}}{|{\bf k}|}e^{i{\bf k}\cdot{\bf
x}}\rangle +c.c.\right] &\equiv& -\sum_{j}\langle [p_j,
Z_j^{\phantom *}-Z_j^{*}] \rangle \nonumber \\ \leq 2\langle {\bf
p}^2 \rangle ^{1/2} \langle -({\bf Z}-{\bf Z^*})^2 \rangle ^{1/2}
&\leq& 2\langle {\bf p}^2 \rangle ^{1/2}\langle2({\bf Z}^*{\bf
Z}+{\bf Z}{\bf Z}^*)\rangle^{1/2}\nonumber\\ &\leq&
\varepsilon\langle {\bf p}^2\rangle
+\frac{2}{\varepsilon}\langle{\bf Z}^*{\bf Z}+{\bf Z}{\bf
Z}^*\rangle.
\end{eqnarray}
Now, each component $Z_j$ can be thought of as a single (unnormalized)
oscillator mode having commutator with its adjoint, $[Z_j^{\phantom
*}, Z_j^*]=(4\pi\alpha /V)\sum_{|{\bf k}|\geq K}k_j^2|{\bf
k}|^{-6}\rightarrow 2\alpha /3\pi K$; moreover, $Z_i^{\phantom *}$ and
$Z_j^*$ commute for $i\neq j$ (i.e., these modes are orthogonal).
Using these facts, we have that
\begin{eqnarray}
\frac{2}{\varepsilon}\langle{\bf Z}^*{\bf Z}+{\bf Z}{\bf Z}^*\rangle
&=& \frac{4}{\varepsilon}\langle {\bf Z}^*{\bf Z}\rangle +
\frac{2}{\varepsilon}\left(\frac{2\alpha}{\pi K}\right)\nonumber \\
&\leq& \sum_{|{\bf k}|\geq K}\langle a_{ \bf k}^*a_{\bf k}^{\phantom
*}\rangle+ 3/2
\end{eqnarray}
if we choose $\varepsilon = 8\alpha /3\pi K $, which is smaller and
better by a factor $1/3$ from the $\varepsilon$ in the article.
Here we have employed an
orthogonal rotation of coordinates bringing $\sum_{|{\bf
k}|\geq K}a_{\bf k}^*a_{\bf k}^{\phantom *} $ into a form
$(4/\varepsilon){\bf Z}^*{\bf Z}+ $non-negative operators. (Compare
Eqs.(21,22) of the article.) Combining these inequalities, we obtain
\begin{equation}
-(\frac{4\pi\alpha}{V})^{1/2}\sum_{|{\bf k}|\geq
K}\left[\langle\frac{{a}_{\bf k}}{|{\bf k}|}e^{i{\bf k}\cdot{\bf
x}}\rangle +c.c.\right] \leq \varepsilon\langle{\bf
p}^2\rangle+\sum_{|{\bf k}|\geq K}\langle a_{\bf k}^*a_{\bf
k}^{\phantom *}\rangle+ 3/2.
\end{equation}
This last inequality is our replacement for the ultraviolet bound
(10). It follows that $H\geq H_K-3/2$ where $H_K$ is as in
Eq.(11), but with the coefficient of ${\bf p}^2$ given by
$(1-8\alpha/3\pi K)$ rather than $(1-8\alpha/\pi K)$. With the
choice $K=
8\alpha/3\pi$, inequality (13) becomes $H\geq
-(16\alpha^2/3\pi^2)-3/2$, a bound at least
consistent with a known {\it upper} bound for the ground state energy
linear in $\alpha$.
The remainder of the article is an analysis of $H_K$ and needs only
minor modification. The coefficient of ${\bf
p}^2$ in Eqs.(19,23,27,28,30) should be $(1-c_1\alpha^{-1/5}/3)$
and, at the end of the article, $c_5= (c_1/3+2c_4)c_P$. Due to the
smaller value of $\varepsilon$ defined above, our estimate
on the
coefficient of $\alpha^{9/5}$ in (31) is slightly improved to 2.337,
rather than 3.822 as reported. Of course, our lower bound for the
ground state energy is decreased merely by the constant $-3/2$, which
is unimportant on a scale of $\alpha^{9/5}$.
\end{document}