From a8e6202c7c35571f29abbdd5ba99226875b94074 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" 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} -- cgit v1.2.3-54-g00ecf