  We continue the systematic study of the thermal partition function of
Jackiw-Teitelboim (JT) gravity started in [arXiv:1911.01659]. We generalize our
analysis to the case of multi-boundary correlators with the help of the
boundary creation operator. We clarify how the Korteweg-de Vries constraints
arise in the presence of multiple boundaries, deriving differential equations
obeyed by the correlators. The differential equations allow us to compute the
genus expansion of the correlators up to any order without ambiguity. We also
formulate a systematic method of calculating the WKB expansion of the
Baker-Akhiezer function and the 't Hooft expansion of the multi-boundary
correlators. This new formalism is much more efficient than our previous method
based on the topological recursion. We further investigate the low temperature
expansion of the two-boundary correlator. We formulate a method of computing it
up to any order and also find a universal form of the two-boundary correlator
in terms of the error function. Using this result we are able to write down the
analytic form of the spectral form factor in JT gravity and show how the ramp
and plateau behavior comes about. We also study the Hartle-Hawking state in the
free boson/fermion representation of the tau-function and discuss how it should
be related to the multi-boundary correlators.
