From a8e6202c7c35571f29abbdd5ba99226875b94074 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Wed, 9 Dec 2020 13:21:55 +0000
Subject: Update on Overleaf.

---
 bezout.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 5fc340a..71c87fd 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -328,12 +328,12 @@ Consider for example the ground-state energy for given $a$, that is, the energy
   }
 \end{figure}
 
-{\color{teal} {\bf somewhere} In Figure \ref{desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
+{\color{teal} {\bf somewhere} In Figure \ref{desert} we show that for $\kappa<1$ there is always a gap of  $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
 For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation
-is more interesting. In the range of  values of $\Re H_0$ where there are real solutions there are solutions
-all the way down to $a=1$: this is only possible if the density of solutions diverges at this value: this is natural, since. 
-
-
+is more interesting. In the range of  values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. 
+Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around
+`deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the $J$'s could produce a real, unusually deep solution for the real problem, in a region where we expect this not to happen.
+}
 \begin{figure}[htpb]\label{desert}
   \centering
   \includegraphics{fig/desert.pdf}
-- 
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