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| -rw-r--r-- | bezout.tex | 35 |
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@@ -164,6 +164,41 @@ not be greater than the product over all singular values \cite{Weyl_1912_Das}. Therefore, the absence of zero eigenvalues implies the absence of zero singular values. +% This is kind of a boring definition... +\begin{equation} \label{eq:count.def.marginal} + \overline{\mathcal N}(\kappa,\epsilon) + =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a) +\end{equation} + +\begin{equation} \label{eq:count.zero.energy} + \overline{\mathcal N}(\kappa,0,a) + =\left[(p-1)a^{p-1}\sqrt{\frac{1-a^{-2}}{a^{2(p-1)}-|\kappa|^2}}\right]^N +\end{equation} + +\begin{equation} + \overline{\mathcal N}(\kappa,\epsilon) + =\lim_{a\to\infty}\overline{\mathcal N}(\kappa,\epsilon,a) + =(p-1)^N +\end{equation} + +For $|\kappa|<1$, +\begin{equation} + \lim_{a\to1}\overline{\mathcal N}(\kappa,\epsilon,a) + =0 +\end{equation} + +\begin{equation} + \lim_{a\to1}\overline{\mathcal N}(1,0,a) + =(p-1)^{N/2} +\end{equation} + +\begin{equation} \label{eq:threshold.energy} + |\epsilon_{\mathrm{th}}|^2 + =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2a^{p-2}} + {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)} +\end{equation} +for $\delta=\kappa a^{-(p-2)}$. + \bibliographystyle{apsrev4-2} \bibliography{bezout} |