Assume that an ordinary integral of the form $$I=\int_{-\infty}^{\infty}dx e^{-f(x)} $$ for some real function $f(x)$ is given where $f(x)$ is well defined over all $\mathbb{R}$ and the integral is convergent. We also assume that $f(x)$ has real and complex saddle points. Now, let us go to complex plane and plot the steepest descent paths (that go through saddles) for $f(z)$ where $z$ is a complex variable. What one usually sees, once done so, is that there is a steepest descent path that lies entirely on the real line and goes through the real saddles. My question is:
Is there any example of a real function $f(x)$ (for which the integral above converges) where the steepest descent path has contributions from the complex saddles? If not, is there any theorem or proof that hints at this issue? In Picard-Lefschetz theory, this integral can be expressed as a sum over the contributions of the Lefschetz thimbles, $$I=\int_{-\infty}^{\infty}dx e^{-f(x)}=\sum_{i=1}^{N} \sigma_i J_i$$
where $\sigma_i$ are the Stokes multipliers of the $i$th Lefschetz thimble contribution $J_i$ and $N$ is the total number of saddles of the function $f(x)$. So in many cases that I am aware of, if $f(x)$ is real, the Stokes multipliers for complex saddle contributions are zero. I want a real example where this is not the case.
Thanks,
AB
