  The Search for Extraterrestrial Intelligence (SETI) has traditionally been
conducted at radio wavelengths, but optical searches are well-motivated and
increasingly feasible due to the growing availability of high-resolution
spectroscopy. We present a data analysis pipeline to search Automated Planet
Finder (APF) spectroscopic observations from the Levy Spectrometer for intense,
persistent, narrow bandwidth optical lasers. We describe the processing of the
spectra, the laser search algorithm, and the results of our laser search on
1983 spectra of 388 stars as part of the Breakthrough Listen search for
technosignatures. We utilize an empirical spectra-matching algorithm called
SpecMatch-Emp to produce residuals between each target spectrum and a set of
best-matching catalog spectra, which provides the basis for a more sensitive
search than previously possible. We verify that SpecMatch-Emp performs well on
APF-Levy spectra by calibrating the stellar properties derived by the algorithm
against the SpecMatch-Emp library and against Gaia catalog values. We leverage
our unique observing strategy, which produces multiple spectra of each target
per night of observing, to increase our detection sensitivity by
programmatically rejecting events which do not persist between observations.
With our laser search algorithm we achieve a sensitivity equivalent to the
ability to detect an 84 kW laser at the median distance of a star in our
dataset (78.5 ly). We present the methodology and vetting of our laser search,
finding no convincing candidates consistent with potential laser emission in
our target sample.
