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| -rw-r--r-- | figs/thimble_stokes_1.pdf | bin | 25662 -> 25662 bytes | |||
| -rw-r--r-- | figs/thimble_stokes_2.pdf | bin | 26978 -> 24421 bytes | |||
| -rw-r--r-- | figs/thimble_stokes_3.pdf | bin | 27074 -> 27074 bytes | |||
| -rw-r--r-- | stokes.tex | 18 |
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diff --git a/figs/thimble_stokes_1.pdf b/figs/thimble_stokes_1.pdf Binary files differindex afc86e1..a3ba1e5 100644 --- a/figs/thimble_stokes_1.pdf +++ b/figs/thimble_stokes_1.pdf diff --git a/figs/thimble_stokes_2.pdf b/figs/thimble_stokes_2.pdf Binary files differindex 78104d7..86f3336 100644 --- a/figs/thimble_stokes_2.pdf +++ b/figs/thimble_stokes_2.pdf diff --git a/figs/thimble_stokes_3.pdf b/figs/thimble_stokes_3.pdf Binary files differindex b06ddcb..0812ec0 100644 --- a/figs/thimble_stokes_3.pdf +++ b/figs/thimble_stokes_3.pdf @@ -14,6 +14,8 @@ \begin{document} +\newcommand\eqref[1]{\eref{#1}} + \title{Analytic continuation over complex landscapes} \author{Jaron Kent-Dobias and Jorge Kurchan} @@ -440,6 +442,22 @@ eigenvalue problem. If we did not know the eigenvalues were real, we could still see it from the second implied equation, $(\beta\partial\partial\mathcal S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. +The effect of changing the phase of $\beta$ is revealed by +\eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and +dividing both sides by $|\beta|e^{i\phi/2}$, one finds +\begin{equation} + \partial\partial\mathcal S(e^{i\phi/2}v) + =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^* + =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^* +\end{equation} +Therefore, one only needs to consider solutions to the eigenproblem for the +action alone, $\partial\partial\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the +resulting vectors by a constant phase corresponding to half the phase of +$\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and +\ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from +left to right produces a clockwise rotation in the thimbles in the +complex-$\theta$ plane. + Something somewhat hidden in the structure of the real hessian but more clear in its complex form is that each eigenvalue comes in a pair, since \begin{equation} |