Wrote paragraph discussing thimble rotation with the phase of β.

1 files changed, 16 insertions, 0 deletions

diff --git a/stokes.tex b/stokes.tex
index 752ba87..0041e38 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -438,6 +438,22 @@ 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}