  The notion of a crosscap state, a special conformal boundary state first
defined in 2d CFT, was recently generalized to 2d massive integrable quantum
field theories and integrable spin chains. It has been shown that the crosscap
states preserve integrability. In this work, we first generalize this notion to
the Lieb-Liniger model, which is a prototype of integrable non-relativistic
many-body systems. We then show that the defined crosscap state preserves
integrability. We derive the exact overlap formula of the crosscap state and
the on-shell Bethe states. As a byproduct, we prove the conjectured overlap
formula for integrable spin chains rigorously by coordinate Bethe ansatz. It
turns out that the overlap formula for both models take the same form as a
ratio of Gaudin-like determinants with a trivial prefactor. Finally we study
quench dynamics of the crosscap state, which turns out to be surprisingly
simple. The stationary density distribution is simply a constant. We also
derive the analytic formula for dynamical correlation functions in the
Tonks-Girardeau limit.
