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| -rw-r--r-- | bezout.tex | 10 |
1 files changed, 5 insertions, 5 deletions
@@ -328,14 +328,14 @@ Consider for example the ground-state energy for given $a$, that is, the energy } \label{fig:complexity} \end{figure} -\textcolor{teal}{ {\bf somewhere} In Figure \ref{fig: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{fig: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} +\begin{figure}[htpb] \centering \includegraphics{fig/desert.pdf} \caption{ |