  In this paper we provide a universal description of the behavior of the basic
operators of the Schwarzian theory in pure states. When the pure states are
energy eigenstates, expectation values of non-extensive operators are thermal.
On the other hand, in coherent pure states, these same operators can exhibit
ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic
or hyperbolic monodromy of an auxiliary equation; or equivalently, which
coadjoint Virasoro orbit the state lies on. These results allow us to establish
an extended version of the eigenstate thermalization hypothesis (ETH) in
theories with a Schwarzian sector. We also elucidate the role of FZZT-type
boundary conditions in the Schwarzian theory, shedding light on the physics of
microstates associated with ZZ branes and FZZT branes in low dimensional
holography.
