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@@ -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} |